I agree, so let me hazard a proposal. We see the nature of establishment/patiṭṭhā operating in AN 3.76 as follows -Precision would also require to specify the nature of 'establishment'.
This sutta suggests that patiṭṭhā is a process of creating the potential for a type of rebecoming in the future. The allowance for both the present event and the future possibility existing in the same proposition is in fact attested to in the special grammatical construction used in "Tasmiṃ patiṭṭhite viññāṇe virūḷhe āyatiṃ punabbhavābhinibbatti hoti." from SN 12.38.Thus kamma is the field, consciousness the seed, and craving the moisture. The consciousness of living beings hindered by ignorance & fettered by craving is established in/tuned to a lower property. Thus there is the production of renewed becoming in the future.
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Iti kho ānanda, kammaṃ khettaṃ, viññāṇaṃ bījaṃ, taṇhā sineho. Avijjānīvaraṇānaṃ sattānaṃ taṇhāsaṃyojanānaṃ hīnāya dhātuyā viññāṇaṃ patiṭṭhitaṃ. Evaṃ āyati punabbhavābhinibbatti hoti.
In this connection, the process of patiṭṭhā seems related to those questions posed to the Buddha regarding the destination/gati of those who have died, eg Ud 1.10, or even of those who have not yet died, eg SN 55.21.
It is this connection to the rebirth potential that causes me to reject the translation of "appatiṭṭhita viññāṇa" as referring to a "consciousness that is unestablished". If such a consciousness exists from the time of the non-establishment all the way to the end of the Arahant's life, we are left with Sati the fisherman's son's belief in some unchanging sub-stratum. I think it therefore makes better sense to treat the term as a reference to the event of non-establishment, rather than as a reference to a special ante-mortem consciousness of the Arahant (or worse, some 'unconditioned' post-mortem consciousness!).
This establishment/patiṭṭhā process looks closely connected to the verb tiṭṭhati which you helpfully pointed out in SN 35.235. I think the only discernible difference I can make out is the fact that patiṭṭhā is a reference to the creation of a future potential, while tiṭṭhati is the present action. Might it be possible to view it sequentially so that "if one tiṭṭhati, then there is patiṭṭhā."?
This split in the respective functions of tiṭṭhati and patiṭṭhāti is in fact explicated by SN 12.38 and SN 12.39 as follows -
The action performed by the verb tiṭṭhati would be the noun ṭhitiyā, while the action performed by the verb patiṭṭhā would be the noun patiṭṭhā . It looks as if ṭhitiyā is the paccaya/condition for the arising of patiṭṭhā.What one intends, what one arranges, and what one obsesses about:[1] This is a support for the stationing of consciousness. There being a support, there is a landing [or: an establishing] of consciousness.
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Yañca bhikkhave, ceteti yañca pakappeti, yañca anuseti, ārammaṇametaṃ hoti viññāṇassa ṭhitiyā. Ārammaṇe sati patiṭṭhā viññāṇassa hoti.
Perhaps we could discuss further in a separate thread.