EDIT: Please see the next response below this one. Having reconsidered the essay, I would alter some of the wording and ideas presented here. I agree that ‘sakkāya’ seems to include all ‘existence,’ in the sense of a rough synonym for ‘bhava’, which includes both the personal incarnation but also the experiences the being has that arise dependent on its rebirth (sense contact, sensation, etc.). In other words, both the internal and external aggregates which condition one another and arise together. I think this also makes a nice connection between sakkāyaditthi, upādāna, and the arising of the aggregates/new life (bhava) that comes through identification. Breaking the shackle of sakkāyaditthi cuts off the fundamental source of the fuel (upādāna) for continued existence, → sakkāya re-arising, because it cuts the delusion underlying identification with the aggregates and taking them up from life to life.
I don’t think ‘kāya’ means ‘substance.’ I think that is a meaning that emerges out of the Jain word ‘astikāya.’ I think that, as in DN 2, ‘kāya’ means a ‘category’ or ‘collection’ of things. The term ‘asti-’ seems to be qualifying that, as saying ‘categories [of things] which truly exist,’ i.e. groups of ‘substances.’ So it seems it is the word ‘asti-’ which allows the sense of ‘substance’ to emerge, while ‘kāya’ is simply a collection.
I think that ‘sakkāya’ is more or less equivalent to a term like ‘attabhāvapaṭilābha.’ Let me explain.
‘Sakkāya’ itself is just defined as the five grasping aggregates. See for example SN 22.105:
And what is sakkāya? It should be said: the five grasping aggregates. …
And what is the origin of sakkāy? It’s the craving that leads to future lives, mixed up with relishing and greed, taking pleasure wherever it lands.
Compare also these two passages:
Mendicants, a Realized One … roars his lion’s roar in the assemblies, and turns the divine wheel.
Such is form, such is the origin of form, such is the ending of form. Such is feeling, such is the origin of feeling, such is the ending of feeling. Such is perception, such is the origin of perception, such is the ending of perception. Such are choices, such is the origin of choices, such is the ending of choices. Such is consciousness, such is the origin of consciousness, such is the ending of consciousness.
SN 12.21
Mendicants, towards evening the lion, king of beasts, emerges from his den, yawns, looks all around the four quarters, and roars his lion’s roar three times. …
In the same way, when a Realized One arises in the world … he teaches the Dhamma: ‘Such is substantial reality, such is the origin of substantial reality, such is the cessation of substantial reality, such is the practice that leads to the cessation of substantial reality.’ …
The Buddha … rolls forth the Wheel of Dhamma from his own insight:
sakkāya, its cessation, the origin of sakkāya, and the noble eightfold path that leads to the stilling of suffering. …
‘We haven’t transcended sakkāya!
It turns out we’re impermanent!’
Note: in both places, the Buddha is said to be like a lion roaring, and turning the Wheel of Dhamma. In one place, it says this is him teaching the four noble truths in terms of the aggregates. In another it is him teaching the four truths in terms of ‘sakkāya.’ Clearly these are parallel, or near identical, teachings. The gods also say that haven’t transcended sakkāya because they are impermanent. This implies ‘sakkāya’ has more the sense of an individual incarnation in the round of rebirths, just as ‘rebirth’ is defined as the ‘manifestation of the aggregates.’
Here, ‘sakkāya’ is given in the exact same framework as the four noble truths, identified as ‘dukkha,’ as something that originates through craving, and is even explicitly defined as the five aggregates! Before we speculate about relations between Jain philosophical terms, we should look at the plain, common, core doctrines of the discourses and the explicit, clear definitions, and see how these ideas fit in there. There is simply nothing here that to my mind indicates that ‘sakkāya’ is a philosophical view of a substantialist reality or ‘truly existent, independent categories of essences.’ That’s how I read passages like the one above for now at least.
As I said in my post above, the suttas do explicitly and clearly say that ‘sakkāya-ditthi’ is the problematic notion or philosophical point that should be given up in the mind. This is consistent with the early discourses, where ‘ditthi’ (views, notions, theories, etc.) are a form of upādāna, a form of ogha, a form of āsava, a form of anusaya, etc. It is standard Early Buddhism that ‘ditthi’ can refer to wrongly held views. So we should classify ‘sakkāya’ under the first noble truth, and we should classify ‘sakkāya-ditthi’ under the second noble truth, as mental defilements that shackle beings to rebirth or the origin of sakkāya.
What is the parallel with ‘attabhāvapaṭilābha’? There are two that come to mind.
- This compound uses a term that would seem to be a point of philosophical debate: ‘attā,’ or ‘self.’ And yet, based on context, the word ‘attā’ is actually not problematized philosophically here. It just refers to a conventional ‘being’ (i.e. ‘oneself’) in a reincarnation.
- The phrase has a derivative of the verb ‘bhavati,’ that is, the -bhāva part, which is similar to ‘sat-’ in ‘sakkāya’ because both relate to ‘existence,’ ‘existing,’ ‘state of being,’ etc.
I would speculate that ‘kāya’ in ‘sakkāya’ does not mean ‘category’ but ‘body.’ Similar to ‘attā,’ which in older literature meant a body or a being. This is true even in Upanisads which advanced notions of a self or ātman; they still sometimes used the word ‘ātman in a more plain sense. ‘Body’ here is not just the physical body, but the whole person, being, or individual. This sense of ‘kāya’ meaning the individual person/incarnation is present in the suttas and is not uncommon. It could also mean ‘category,’ in the sense of the categories (synonym of ‘khandha,’ aggregate) that make up a concrete existence or state of being (an incarnation).
The prefix ‘sat- I would speculate is a general intensifier that points to a ‘real/whole being,’ i.e. a concrete state of being with a particular body/incarnation. Similar to ‘attā,’ I don’t think ‘sat-’ is taking on a problematic philosophical sense here. This is not to deny that that connotation lies dormant in the word; it certainly seems like it does, and that is related to the use of the term with ‘ditthi.’ But it seems to me that ‘ditthi’ is the notions or views that arise based on and around a concrete individual existence, i.e. which take it to be ‘concrete’ or ‘existent’ or ‘individual’ in a more literal, philosophical sense. Please keep in mind by ‘individual existence’ I do not mean in the sense of identifying with the conceit ‘I am.’ It would apply equally to a living arahant or puthujjana, as both are manifestations of a ‘concrete existence.’ The former just does not have a ‘view about concrete existence,’ i.e. that is their self or that it is substantial.
This sense of the word also seems to be how it was used in some Buddhist contexts that are slightly later. This is from the dictionary entry at wisdomlib:
mā…imaṃ…satkāyaṃ kāyaṃ manyadhvam (Aṣṭasāhasrikā-prajñāpāramitā 94.12), don’t think this existent (physical) body (of Tathāgatas) is their body (but rather regard the dharmakāya as such)
Here, ‘sakkāya’ clearly just means the concrete incarnation that is present in existence, and it refers to the Buddha (i.e. an arahant).
Bhikkhu Bodhi similarly understands ‘sakkāya’ to to be the ‘basis’ for the arising of wrong views, not as implying the wrong view itself. In his recent work, Reading the Buddha’s Discourses in Pāli, he says:
The aggregates are the ‘personal-collection’ (sakkāya), the objective basis of the view [of sakkāyaditthi], and the view is the notion that there is a self existing in some specific relation to the aggregates.
This is at least some of my speculation and inclination for now. Please correct any mistakes in my reading! 
