If I understand correctly, the Rav doesn't believe that God is very involved in this world—so why does he pray to Him if He's not involved?

I looked at a couple of your posts. If I understand correctly, your answer seems to suggest the opposite—that it's not really going to help. So why do we pray? Why do we ask Hashem to cure us (Refa'einu) or provide us with livelihood (Barech Aleinu)? I'm specifically referring to the Shemoneh Esrei (Amidah) prayer

See, for example, column 280 and the end of 298. You can search the site for the meaning of prayer in general and for requests in particular.

I was reading but I don't feel like you are answering this question

I definitely answered there and in other places. I said that sporadic intervention by God cannot be ruled out and therefore there is no justification for changing the prayer (something that one minyan would need another minyan to allow). Therefore, one should focus on those who are in a situation without a natural solution and hope that perhaps sometimes they will be answered. In any case, I would not pray beyond the regular prayer except in desperate cases.

Beyond the request, there is praise and confession, which I have explained. Praise is not problematic at all, and confessions about a miracle are simply an opportunity for us to give thanks for the world and everything in it.