You have the circuit figured out correctly. (Basically the same as Chevy, except for ammeter and a few other things.)

I hate fixing electrical stuff. Usually I find the wires of the affected circuit (which you have done) and just start replacing things. (I have replaced headlight switch, turn signal switch, and flasher.) My brake lights did not work and my notes say it was a wiring problem (I did not buy any parts)  but it was so long ago (2008) I don't remember exactly what it was. Might have had to do with the fuse block and wires connected there, since I have no less than 7 detailed pictures of it in my files.

Since this happens after you have been driving a short while, maybe something is affected by movement? What seems most likely in that case is the brake light switch - try replacing that, get the one for the Chevy (NOT Pontiac), not hard to find, cheap, easy to replace (Advance Auto Parts has them in stock at my local store, $6.99 US).

Since turn signals still work, not likely that the turn signal switch is the problem - but pull the horn button, inspect the switch carefully, especially the white wire coming in, is it firmly affixed to the switch? Frayed where it comes out of the column toward the switch? Also check at the bottom of the steering column where that miserable (if you ever have to replace this wiring, you will find out why I call it miserable) 6-strand wire bundle comes out, any fraying of the wires?

How about the red wire that comes into the fuse block - connection good, any fraying of the wire?

Hope this helps.

Dave