Hi KYG. I agree with you above. It seems the contextual uses of avakkanti & okkamati are so diverse in the suttas that I think our consideration of these words must be extremely broad and flexible.
The use of the various words in the various suttas you quoted is very interesting for me; even the use of similar words such as ‘virūḷhaṁ’ & ‘vuddhi’.
In SN 12.64, SN 12.59 and SN 22.47, there seems to be an arising occurring before the avakkanti.
- In SN 12.64, desire, relishing and craving have arisen before the patiṭṭhitaṁ & virūḷhaṁ of consciousness and the avakkanti of nama-rupa.
- In SN 12.59, there has arisen dwelling pondering/contemplating the gratification about things that fetter (saṁyojaniyesu dhammesu assādānupassino viharato) before the avakkanti of consciousness.
- In SN 22.47, the conceit “I” is arising before the avakkanti of the five sense faculties.
Based on my limited reading, at this time, I think any inference that patiṭṭhitaṁ (from SN 12.64) is synonymous with avakkanti should be avoided.
Patiṭṭhitaṁ is particularly notable in SN 22.53, where it seems to simply refer to consciousness becoming established/fixed with delight (nandi) upon certain sense objects. Also, both vuddhiṁ (grow) & virūḷhiṁ (increase) are used in SN 22.53. Therefore, in SN 12.64 & SN 22.53, first the consciousness becomes ‘established’ (‘patiṭṭhitaṁ’) and then it proliferates (grows & increases). Then from this, there is the avakkanti of nama-rupa.
SN 12.59 seems similar because it seems the consciousness is already established in a gratifying fettering object via the use of the words “ānupassino viharato”. Although the word ‘patiṭṭhitaṁ’ is not found here, “ānupassino viharato” (“dwelling contemplating”) has a similar meaning. Therefore, in SN 12.59, there is dwelling contemplating the delightful object; then from this there is the avakkanti of consciousness.
In SN 12.64, after the patiṭṭhitaṁ of consciousness & avakkanti of nama-rupa there is the vuddhi of sankhara. The vuddhi here is similar to the same word (vaddha) in MN 149, which says:
Mendicants, when you don’t truly know and see the eye, sights, eye consciousness, eye contact, and what is felt as pleasant, painful, or neutral that arises conditioned by eye contact, you’re aroused by desire for these things.
“Cakkhuṁ, bhikkhave, ajānaṁ apassaṁ yathābhūtaṁ, rūpe ajānaṁ apassaṁ yathābhūtaṁ, cakkhuviññāṇaṁ ajānaṁ apassaṁ yathābhūtaṁ, cakkhusamphassaṁ ajānaṁ apassaṁ yathābhūtaṁ, yamidaṁ cakkhusamphassapaccayā uppajjati vedayitaṁ sukhaṁ vā dukkhaṁ vā adukkhamasukhaṁ vā tampi ajānaṁ apassaṁ yathābhūtaṁ, cakkhusmiṁ sārajjati, rūpesu sārajjati, cakkhuviññāṇe sārajjati, cakkhusamphasse sārajjati, yamidaṁ cakkhusamphassapaccayā uppajjati vedayitaṁ sukhaṁ vā dukkhaṁ vā adukkhamasukhaṁ vā tasmimpi sārajjati.
Someone who lives aroused like this—fettered, confused, concentrating on gratification—accumulates the five grasping aggregates for themselves in the future.
Tassa sārattassa saṁyuttassa sammūḷhassa assādānupassino viharato āyatiṁ pañcupādānakkhandhā upacayaṁ gacchanti.
And their craving—which leads to future rebirth, mixed up with relishing and greed, looking for enjoyment in various different realms—grows.
Taṇhā cassa ponobbhavikā nandīrāgasahagatā tatratatrābhinandinī, sā cassa pavaḍḍhati.
Their physical and mental stress,
Tassa kāyikāpi darathā pavaḍḍhanti, cetasikāpi darathā pavaḍḍhanti;
torment,kāyikāpi santāpā pavaḍḍhanti, cetasikāpi santāpā pavaḍḍhanti;
and fever grow.
kāyikāpi pariḷāhā pavaḍḍhanti, cetasikāpi pariḷāhā pavaḍḍhanti.
And they experience physical and mental suffering.
So kāyadukkhampi cetodukkhampi paṭisaṁvedeti.
MN 149
My conclusions: … START AGAIN 
SN 12.64 is similar to the Dukkhavagga, which contains the following suttas:
- SN 12.51; a questionable sutta containing Abhidhamma principles. We can ignore SN 12.51.
- SN 12.52, which is about there are things that are prone to being grasped. When you concentrate on the gratification provided by these things, your craving grows.
- SN 12.53, which is about there are things that are prone to being fettered. When you concentrate on the gratification provided by these things, your craving grows.
- SN 12.54, which is about there are things that are prone to being fettered. When you concentrate on the gratification provided by these things, your craving grows.
- SN 12.55, which is about there are things that are prone to being grasped. When you concentrate on the gratification provided by these things, your craving grows.
- SN 12.56, which is about there are things that are prone to being grasped. When you concentrate on the gratification provided by these things, your craving grows.
- SN 12.57, which is about there are things that are prone to being fettered. When you concentrate on the gratification provided by these things, your craving grows.
- SN 12.58, which is about there are things that are prone to being fettered. When you concentrate on the gratification provided by these things, name and form are conceived.
- SN 12.59, which is about there are things that are prone to being fettered. When you concentrate on the gratification provided by these things, consciousness is conceived.
- SN 12.60, which is about there are things that are prone to being grasped. When you concentrate on the gratification provided by these things, your craving grows.
SN 12.63, which is in the Mahāvagga, is the first sutta about the Four Nutriments, explaining what the Four Nutriments are.
SN 12.64, is the next sutta, applying the principles in the Dukkhavagga to the Four Nutriments. SN 12.64 says if there is desire, relishing and craving for the Four Nutriments of solid food, contact, mental intention &/or consciousness, then consciousness becomes established there and grows (patiṭṭhitaṁ virūḷhaṁ); name and form are conceived (avakkanti) and there is the growth (vuddhi) of mental formations/proliferation (sankhara).
In conclusion, about SN 12.64, since consciousness as one of the Four Nutriments is potentially an object of craving, here, it seems what is subject to avakkanti is nama-rupa because, similar to solid food, as a Nutriment, consciousness is something a Buddha must also use. For example, MN 43 says wisdom & consciousness are co-joined therefore there cannot be wisdom without consciousness. In other words, in SN 12.64, it seems consciousness is treated as something ‘neutral’ rather than as inherently unwholesome. So in SN 12.64, when the problem of defilement emerges (avakkanti), it is because of nama-rupa. In other words, nama-rupa is giving inappropriate attention to the Four Nutriments (rather than using the Four Nutriments with wisdom).
If you do not understand what I am concluding above, we must return to SN 12.63, which is difficult yet also simple to understand. SN 12.63 points out the problem of defilement (in bold text) that can occur (when there is ignorance) in relation to the Four Nutriments, as follows:
- When solid food is completely understood, desire [rago/lust] for the five kinds of sensual stimulation is completely understood.
- When contact as fuel is completely understood, the three feelings are completely understood.
- When mental intention as fuel is completely understood, the three cravings are completely understood.
- When consciousness as fuel is completely understood, name and form is completely understood. When name and form are completely understood, a noble disciple has nothing further to do, I say.
In summary, SN 12.63 instructs:
- When solid food must be eaten, eat it like you are eating your own son’s flesh. Do not eat it with lust. Just eat enough for the journey across the desert.
- When contact must occur, be mindful to endure the “biting/nibbling” of feelings, so to not generate craving.
- When intention must be used, ensure the intention is Noble Intention rather than craving.
- When consciousness is experienced, ensure nama-rupa does not generate wrong intention & inappropriate attention towards consciousness. Keep in mind SN 12.2 defines nama-rupa as feeling, perception, intention, contact & attention.
Therefore, to conclude about SN 12.64, my personal theory/working hypothesis 
is the avakkanti of nama-rupa occurs because nama-rupa is the potentially problematic active dhamma here (because consciousness is passively rendered as one of the sense objects/nutriments). 
Therefore:
- In SN 12.58, an unskillful problem (of assādānupassino viharato) arises at nama-rupa when you concentrate on the gratification provided by things. Thus nama-rupa is avakkanti.
- In SN 12.59, an unskillful problem (of assādānupassino viharato) arises at consciousness when you concentrate on the gratification provided by things. Thus consciousness is avakkanti.
- In SN 22.47, an unskillful problem arises at the five sense faculties when the conceit “I am” is not abandoned. Thus the five sense faculties is avakkanti.
Assuming, for now, the above hypothesis is correct, namely, avakkanti is applied to the primary active emergent dhamma, we can now consider other uses of these words:
- MN 121 - That’s how emptiness is born [emerges; enters] in them—genuine, undistorted and pure. Evampissa esā, ānanda, yathābhuccā avipallatthā parisuddhā paramānuttarā suññatāvakkanti bhavati.
- DN 15 - If consciousness were not conceived [entered] in the mother’s womb, would name and form coagulate there? Viññāṇañca hi, ānanda, mātukucchismiṁ na okkamissatha, api nu kho nāmarūpaṁ mātukucchismiṁ samuccissathā”ti?
- AN 3.22 - Some people can enter the sure path with regards to skillful qualities, but only if they get to see a Realized One, and to hear the teaching and training that he proclaims, and not when they don’t get those things. Idha pana, bhikkhave, ekacco puggalo labhantova tathāgataṁ dassanāya no alabhanto, labhantova tathāgatappaveditaṁ dhammavinayaṁ savanāya no alabhanto okkamati niyāmaṁ kusalesu dhammesu sammattaṁ.
- SN 12.2 - The rebirth, inception, conception [emergence; entry], reincarnation, manifestation of the aggregates, and acquisition of the sense fields of the various sentient beings in the various orders of sentient beings. Yā tesaṁ tesaṁ sattānaṁ tamhi tamhi sattanikāye jāti sañjāti okkanti abhinibbatti khandhānaṁ pātubhāvo āyatanānaṁ paṭilābho.
- AN 5.158 - “Mendicants, a mendicant with five qualities is overcome by timidity. Pañcahi, bhikkhave, dhammehi samannāgato bhikkhu sārajjaṁ okkanto hoti.
- AN 7.61 - So, Moggallāna, don’t focus on or cultivate the perception that you were meditating on when you fell drowsy. Tasmātiha, moggallāna, yathāsaññissa te viharato taṁ middhaṁ okkamati, taṁ saññaṁ mā manasākāsi, taṁ saññaṁ mā bahulamakāsi.
- Iti 83 - Their flower-garlands wither; their clothes become soiled; they sweat from the armpits; their physical appearance [falls into; enters into deterioration] deteriorates; and they no longer delight in their heavenly throne. mālā milāyanti, vatthāni kilissanti, kacchehi sedā muccanti, kāye dubbaṇṇiyaṁ okkamati, sake devo devāsane nābhiramatīti.
- Snp 2.7 - This unnatural violence has been passed down [descended from; emerged from] as an ancient custom. Okkanto purāṇo ahu;
- SN 51.22 - Sometimes the Realized One submerges his body in his mind and his mind in his body. He meditates after sinking [entering] into a perception of bliss and lightness in the body. Yasmiṁ, ānanda, samaye tathāgato kāyampi citte samodahati cittampi kāye samodahati, sukhasaññañca lahusaññañca kāye okkamitvā viharati;
- AN 3.61 Supported by the six elements, an embryo is conceived [emerges; develops; enters]. Channaṁ, bhikkhave, dhātūnaṁ upādāya gabbhass āvakkanti hoti;
Therefore, for now, I maintain my hypothesis that ‘avakkanti’ is used for the primary active dhamma (whether wholesome or unwholesome) that is emerging/emergent (avakkanti). It follows, in SN 12.64, consciousness is not the primary active dhamma but instead is a secondary passive dhamma. The primary active dhamma in SN 12.64 is namarupa.
Note: for me, SN 12.58, SN 12.59 & SN 12.64 are not related to DN 15 or AN 3.61. DN 15 seems certainly about consciousness entering into the mother’s womb while SN 12.58, SN 12.59 & SN 12.64 seem about the consciousness & namarupa in relation to sense objects.