Pītiyā ca virāgā upekkhako ca vihāsim, sato ca sampajāno sukhañca kāyena paṭisamvedesim; yam tam ariyā ācikkhanti: ‘upekkhako satimā sukhavihārī’ti
"Why do we personally think sati sampajāno is mentioned?"
I personally think in this context sampajāno is used to specify and clarify the exact meaning in which sati is used here.
I think this is evident from the subsequent phrase "upekkhako satimā sukhavihārī’ti" — a (then) popular saying that uses sati in the sense of sampajāna but does not explicitly mention sampajāna.
In general, I agree with and support your interpretation that in many contexts sati means remembrance of the instructions or, even more precisely, having one's awareness be positioned and settled within the framework given by the instructions.
But, evidently, neither this particular passage nor this sutta at large set any context, expressed or implied, of following the instructions or remembering the instructions.
Instead, in this passage the Buddha explains what he means by dwelling in equanimity; first he simply describes it as a kind of mindfulness (sati) and then he clarifies that he means the peaceful bliss of being present and aware in the here and now. To this end he quotes the popular saying that succinctly summarizes what he just described.
From the rest of the jhana sequence we know this dwelling in equanimity, the peaceful bliss of being present and aware comes from having no cravings and no aversions, not making vitakka/vicara effort, no revulsion and no joy that would come from one judging phenomena as agreeable or disagreeable. The only remaining asava at this point are the subtle aversion to dukkha and the subtle attachment to peace inherent in this jhana itself, to be overcome in the next jhana.
This is what I personally think.