Even without the damage from the rain, the bridge set would have been a decade (at least, I'm not sure if any of it was built for Phase 2) old by the time of Star Trek V
and had never been intended for long term use, chances are it would have needed something of a major revamp and refresh even if it had survived. Espcially factoring in the advances in computer and video technology over the 80's.
To be honest, I wonder if it was starting to look a little worn before that, after the return to Earth at the start of III whenever we see the bridge it tends to either have very low lighting (a side affect of having to steal it seems to be they've set the cinematography to moody even before the Klingon's attack, plus it's use as the Saratoga in IV is all dark as well) or very carefully framed as the initial A bridge (where they could only afford/had the time to jazz up the bits you see on screen).
oddly, I think it's use as the Grissom is the last time we see the original bridge set in all it's brightly lit glory on the big screen. And even it's TNG reuses tend to keep the lighting much lower than was normal for that series (the difference between the battle and regular bridges on the D is quite striking).