SuttaCentral
So the translation would need to describe insight into the correct and unified Dhamma via meditation.
Does Meditative insight of The Dhamma or Mindful insight of The Dhamma convey your point? It seems to fulfill Ven Analayo’s and Ven Bodhi’s statements you quoted in your essay: meditative insight which is penetrative, thorough and unified with the truth of the Buddha’s teaching.
manasikāra: application of mind
sammāmanasikāra: right application of mind
IMHO, these renderings lead naturally to:
yonisomanasikāra: originary application of mind
originary (rare): of, pertaining to, or causing the origin/source of something (else)
“application of mind” seems so clunky to me; however, if we accept it as the rendering of manasikāra, then this proposal seems ok.
Here are three way to consider “yoniso manasikāra” as directing the mind in accord with a fundamental matrix. (1) if a matrix is a medium containing objects, like gems contained in the earth, then yoniso manisikara would make us not be distracted by the shiny object, but instead pay attention to the entire picture. Then we would look at the universal lakkhana of existence. (2) if a matrix is the underlying order from which specific phenomena arise like the genetic code for a plant or animal. Then we would look at causation. (3) if the matrix is the environment in which something develops, then we would consider the entire wings to awakening as the matrix for the arising of wisdom. [Thank you all for this conversation!]
Might be a pre-vedic term even? Wikipedia on ‘yoni’:
The reverence for yoni, state Jones and Ryan, is probably pre-Vedic. Figurines recovered from Zhob valley and dated to the 4th millennium BCE show pronounced breasts and yoni, and these may have been fertility symbols used in prehistoric times that ultimately evolved into later spiritual symbols.[6] According to David Lemming, the yoni worship tradition dates to the pre-Vedic period, over the 4000 BCE to 1000 BCE period.[29]
Isn’t this a little more phallicus manasi kāra than yoni? :3
Why can’t it mean to feel deeply and/or intuitively? Emotions and intuition presently have low status because they are associated with the feminine. Was that the case in the time of the Buddha – or earlier – as well? The brahmaviharas are very feely-feely, but no one seems to look down on them in the EBTs 
'+ the Dhamma is said to be beyond mere reasoning…
Nope. As per the OP, there are basically three sense of yoni:
This makes sense of the commentarial gloss upāyena (“by method”) which is incomprehensible if we only take yoni in the third sense. This sense is, IMHO, fully justified in the suttas which treat it as a means or path by which one can trace back to the cause. The normal word in English for a step by step method of tracing back to the cause of things is “rationality”. Both “fundamentally” and “radically” lean on the third meaning.
Acceptable alternatives would be things like “systematic” or “methodical”.
It can, but is not limited to that. May I gently suggest that modern Buddhism, and especially modern American Buddhism, is tainted by an anti-rationalism that is not Buddhist and needs to be confronted?
Just yesterday I listened to a talk between Jack Kornfield and Sam Altman on AI. I’ve never heard Jack speak before. He quoted Nisargadatta saying “the head creates the abyss, the heart heals it” or something like this. This kind of head/heart or intellect/emotion dichotomy is completely alien to Buddhism. What creates the abyss is greed, hate, and delusion, of which greed and hate are most definitely matters of the heart.
Don’t confuse
with
It’s the ism that’s the problem!
It could, but as I explained at length in the OP, that is not what yoni actually means. It means the path or passage or means by which one moves towards the truth of origination.
Thanks for noticing! But I have to admit, the OP is not very clear or coherent. I’ve quickly edited it a bit and added a warning. But it basically grew and changed as i learned more so it could be redone far more clearly (and I hope persuasively!)
Not really, it’s too obscure, and once again leans too heavily on sense three above. Yoni here doesn’t cause the origin of something, it moves towards the discovery of it.
These are all nice senses, I just don’t think they are what the word actually means. One problem with all three explanations here is that they are all locative: the matrix in which the events happen. But the term is instrumental, the manner by which something proceeds.
Sure, fertility worship is very old and widespread. But it seems that the word itself is from a Proto Indo European root, and interestingly enough, that root supports my conclusion. According to Pokorny, the original sense is “way”, “right way”.
https://indo-european.info/indoeuropean.html
Ha ha, no, if anything vaginicus maniskāra: the path or way or means. Specifically, the means by which the soma reaches the cup, which symbolically represents the means by which the semen reaches the uterus.
I was not pointing to emotional intuitions as an alternative to logic, but rather to what can be known directly pajānāti as the safeguard against going off into a wrong view based on an incorrect or incomplete premise. Kalupahana in Buddhist Philosophy a historical analysis relies on this distinction. As i just wrote elsewhere, for me personally, yoniso manasikāra is pointing to the fundamental nature of reality, to causation, and to the systematic Dhamma which is the matrix for the arising of wisdom.
Sure, but what can be known directly? Sights, sounds, smells, tastes, touches, and present mental phenomena. That’s it.
Not impermanence: that is an inference. Not the four noble truths: that’s another inference.
To know directly (paccakkha) is not the same thing as to know truly (yathābhūta).
So the arising of vijjā is happening in context of both concepts and things that can be known or experienced directly.
Indeed, yes. That both direct kowledge and inference are the two aspects of true knowledge is a common axiom of all Buddhist (and probably all Indic) forms of epistemology. I believe that this only stopped being the case in the 20th century, under the influence of people like Krishnamurti, DT Suzuki, etc.
AN10.61:6.20: Ko cāhāro yonisomanasikārassa?
And what is the fuel for yonisomanasikārassa?
AN10.61:6.21: ‘Saddhā’tissa vacanīyaṁ.
You should say: ‘Faith.’
If manasikārassa is considered ‘ideation’, and fuel for yonisomanasikārassa is faith, then yonisomanasikārassa could be considered ideation by faith. By faith in what? By faith in true teaching, God’s Word, Truth.
Heb 11:1 Now faith is the substance of things hoped for, the evidence of things not seen.
However, if one consider some other translation, eg: ‘rational application’: Then by using AN10.61:6.21, we can ask the questions to test the translation: Is faith the source of rational application? Can there be rational application of mind without faith?
If i read the translations of Bodhi many things can and must be directly known. A collection:
MN1.50. "He directly knows Nibbana as Nibbana. Having directly known Nibbana as Nibbana, he should not conceive [himself as] Nibbana, he should not conceive [himself] in Nibbana, he should not conceive [himself apart] from Nibbana, he should not conceive Nibbana to be 'mine/ he should not delight in Nibbana. Why is that? So that he may fully understand it, I say. (Bodhi)
MN1 also says that elements are directly known.
One can also directly know the deathless (AN1.625)
One can also directly know …gratification in the world, danger and escape of this world (AN3.105)
Just as some examples in the texts.
In note 1397 of Bodhi’s translation of AN, he refers to the commentary which says: Mp: "What should be directly known (abhihneyyam) is the four noble truths.
I think there can be, or at least, there is the semblance of it, but genuine inquiry always takes places in the presence of faith, which warms and heartens the mind in its journey into the unknown.
If we take “yoniso” as “systematic”/“methodical”, how would you translate “ayoniso”?
“Unmethodical”, I guess?
antonyms for methodical in the thesaurus: haphazard, disorganized, hit-or-miss, unsystematic. These words convey why one would look down on ayoniso manisikāra!
This may not be so useful to the discussion.
I’ve checked our Thai dictionary.
โยนี
น. อวัยวะสืบพันธุ์ของหญิง (Pali, Sanskit มดลูก; ที่เกิด, ต้นกำเนิด; ปัญญา)
= Uterus; place of birth, origin; wisdom
The third definition is the most curious, IMHO.
Do you find ‘origin’ surprising?
For me, the fourth, ‘wisdom’ seems a bit more, although it seems to line up with PED’s #3:
which comes after the more literal ‘womb’, ‘place of birth’, ‘plane of existence’, etc.
The third on is the ‘wisdom’. The semicolon is a bit difficult to see. 
Ah, yes. So we think alike!
I suppose that would be the implied meaning of yoniso manasikara: ‘wise attention’.