Both gītassarena and tantissarena are clearly referring to music - the word tanti (sanskrit: tantrī) referring to a “stringed” lute instrument such as the vīṇā which are normally used in classical music performances.
Indian classical music is svara-based and rāga-based music - where svara means a solfa-note representing a particular pitch. It is said that this type of music originates in the musical-type enunciation of the sāmaveda which is sung using 7 notes (as in music) rather than chanted. In any case, the svaras of music and the svaras of the vedas represent nearly the same thing conceptually -i.e. pitch-based tones. The Vedic language was itself pitch-based, and the classical music tradition where instruments and singing were used were also svara-based. Singing normally uses many more svaras than chanting, though - and the function of a pitch in music differs from chanting.
The pitch-based tones of the vedas are a semantically-relevant linguistic feature i.e. which could change the meanings of the words used (a feature also shared by other ancient Indo-European languages like Ancient-Greek, Proto-Germanic (the ancestor of English) etc, but the pitch-based tones of music are purely a musical feature which do not impact semantics - the musical-svaras probably originate conceptually from the linguistic-svaras of Vedic, and the two are related to each other, but are not identical in usage. You can hear the 7 svaras of classical Indian music at https://www.youtube.com/watch?v=uDd3iupKUyI (between 0:40 and 1:10).
That such a svara-based form of classical music tradition did exist in the vedic period is also evidenced in Vedic texts. This is an extract from the Gopatha Brāhmaṇa of the Atharva-veda:
om iti vyāhṛtīḥ svaraśamyanānātantrīḥ svaranṛtyagītavāditrāṇy anvabhavac…
In the above phrase you can see the words svara (pāli: sara), tantrī (pāli: tanti), gīta (pāli: gīta), nṛtya (pāli: nacca) etc where the discussion is about music and dancing.
The language of the other vedas (apart from the sāmaveda) and classical sanskrit, on the other hand - have only 3 svaras (udātta, anudātta and svarita) - and these 3 svaras are what Pāṇini (in circa the 4th century BCE) says are used in spoken classical sanskrit. Paninian grammar explains the svara system of classical sanskrit (and of vedic) in great detail and the evidence both from Pāṇini and from other literature from the time indicates that the svara enunciation in classical sanskrit was still in use (but not invariably used like in Vedic) in the 4th century BCE.
The svara enunciation in classical sanskrit has now gone out of use in speech, but in the 4th century BCE it was still employed optionally. There is no hard and fast rule demarcating classical sanskrit from vedic sanskrit, and vedic grammatical and other archaisms did continue in classical sanskrit even after Pāṇini (for example Patañjali, in his 2nd century BCE commentary on Pāṇini’s grammar, the Vyākaraṇa-Mahābhāṣya, written in classical sanskrit, uses the vedic ‘tavai’ infinitive affix – “tasmād brāhmaṇena na mlecchitavai nāpabhāṣitavai”).
Many other examples of the continuation of Vedic grammatical, phonetic & linguistic usages are available in a lot of BCE texts, including in the Pāli canon. So classical sanskrit in the time of Pāṇini is a broad term that simply means late-Vedic (as spoken in the 4th century BCE), it does not mean that it had rid itself of all grammatical/phonemic/lexical archaisms that characterise Vedic.
I agree there are no svaras in Pali, neither are there any svaras in any of the other prakrits.
However, chanting the aṭṭhakavagga with svara would not be the same as singing it, if it were singing, it would be described as singing (gītam), also singing would have involved a lot more than simply using svaras i.e. it would probably have had to be set to a musical mode (rāga) first. Setting the aṭṭhakavagga to music is not evident to me from the quoted phrases.
Besides the Divyāvadāna also evidences the intoned recitation that I have postulated above – see “athāyuṣmāñ śroṇo bhagavatā kṛtāvakāśaḥ asmāt parāntikayā guptikayā udānāt pārāyaṇāt satyadṛṣṭaḥ śailagāthā munigāthā arthavargīyāṇi ca sūtrāṇi vistareṇa svareṇa svādhyāyaṃ karoti” - here, apart from the word svareṇa, the usage of the word svādhyāya is also significant. Svādhyāya in the vedic tradition refers to chanting the memorized vedic hymns periodically. Each brahmin clan is associated with a specific set of vedic verses handed down in an oral patrilineal tradition that goes back to the original vedic Ṛṣis who composed those verses. So svādhyāya means they periodically recite the memorized set of hymns that they had inherited from their ancestors with an intention to pass it down the line. In the buddhist sense, the above statement in the Divyāvadāna (and in the Pāli canon where too sajjhāya is mentioned as being done both by brahmins and buddhists of the buddha’s lifetime) would mean not an oral transmission across generations (unlike the vedas) but chanting the suttas of the arthavargīyāṇi (aṭṭhakavagga), śailagāthā (sela-sutta), the munigāthā, the satyadṛṣṭa, the verses of the pārāyaṇavarga, udānas etc as these early suttas would have been regarded as foundational texts of the buddhist canon, and memorizing / reciting them with their original accents was evidently considered a form of svādhyāya.
Yes it was a distinguishing feature, but only in late-vedic. A lot of people didnt or couldnt speak intoned sanskrit (they spoke non-accented sanskrit which has since, become the invariable norm), but evidently still understood intoned speech when spoken to – back then – as some others could and did speak the language with accents. The svaras when employed incorrectly changed the meanings of words (expressed contrarian senses), and examples of such semantic shifts occuring as a result of incorrect use of the accents are discussed in sanskrit grammatical texts.
However since the Sanskrit tradition (from which the Divyavadana quotes) also refers to such an intoned recitation, the issue is not unique to Pāli.