Surely, when shooting driven game, the most distant targets demanding the tighter choke are tackled first. In these circumstances, do I pull the back trigger first, or am I missing something?
You are right in that most side-by-sides fire the barrel with the most open choke via the front trigger.
You are also right in that, when shooting driven game, the first target is usually the most distant, demanding the tighter choke of the pair.
However, in walked-up shooting the reverse is true: the first target is usually the nearer of the pair.
You are not missing anything, and the only solution is, when shooting driven game and the circumstances demand, to pull the back trigger first.
This is where over-unders score over side-by-sides: you can set the barrel selector to fire the tightest choke first.