Poor choice of words on my part. I didn't mean that the entire reactor shut down, just that there may have been some way to isolate the dilithium circuit for safe inspection.
As to how important the DCC is, in Elann of Troyius
the lack of a functioning crystal is disasterous:
SCOTT: I've got bad news, Captain. The entire dilithium crystal converter assembly is fused. No chance of repair. It's completely unusable.
KIRK: No chance of restoring warp drive?
SCOTT [OC]: Not without dilithium crystals. We can't even generate enough power to fire our weapons.
Of course other sabotage had also gone on, so it may well be that under ideal conditions routine inspection (and replacement) of the crystal while the enfines are running is no big deal.