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.
If you check out a copy of Shane Johnson's Mr. Scott's Guide to the Enterprise
(written during the production of TVH, predating TFF), there is an excellent section on the Enterprise
-A, and particularly its bridge, with in-universe flavor-text detailing the "new" console-technology employed aboard the NCC-1701-A that was different from its predecessor -- touchpad controls, more advanced viewscreen technology, etc.
There are a number of close-up stills of the Voyage Home
bridge, which makes for an interesting contrast with how the bridge evolved between the productions of TVH and TFF.
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).
Only the "rear" arc of the bridge (meaning from the Science station, to the turbolift, and over to Uhura's console, plus the helm and the captain's chair) were actually built for TVH, in order to save money -- the "forward"-facing arc of the bridge was left unmade, with no viewscreen constructed for the Ent
-A until Herman Zimmerman did so for TFF.