My aim is similar to yours. I took up piano late after playing saxand other instruments.

I once asked my sax teacher what chords to learn first, he just said 'them all, your going to need them all'.

When you think about it, if yo were teaching a math student multiplication tables and they asked the same question, the answer would be the same.

I had a bass player visit my studio the other day, a pro. He got out the Real Book sight read a few songs, first recording a bass part, then recorded a harmony track (all on the bass!) then a melody track - using bass again (upper register). He did this without even thinking about it, without making errors and without breaking a sweat.
Playing with bands (on sax) I remember being laughed at for asking the key (with good players) I was expected to hear it and jump in. Baptism by fire, I know, and there are other bands that would not do this.

The key to all this is to keep it simple, learn something within your grasp, but something you can use as a building block in a methodical way. Always play with feel and accuracy - this is where BIAB is so useful. I use it three to eight hours a day, I want to achieve freedom on the piano, I am in my seventh year. I do play tunes, I memorise them all. This hampers my sight reading for the piano, but I dont really need this except for Real Book level. I find that it also pays to work on a tune down to its very core, not only the straight melody as played, but also lots of variations based on theory. I learn a lot this way.

When I was young I played trumpet and cornet. I used to sight read everything and I got really fast at it. I remember doing all of the Beethoven Symphonies and lots of the competition pieces for Brass bands, however I could not play a note without a sheet of paper. I always wanted to play freely, from the heart, without barriers. None of my Band members (well respected players - some of them world cornet champions brass band style) could improvise. Its a different culture, the classical training, I believe it robs the player of creativity.
On sax, my reading is mediocre (and jazz is harder to read than classical because of syncopation) but my impro skills are fine now. I play hundreds of standards.

Z

Win 11 64, Asus Rog Strix z390 mobo, 64 gig RAM, 8700k