It could even add to the sense of drama, in that here is a problem so serious it requires going past the speed limit.
That's exactly the issue. The Enterprise is always going to be involved in emergencies and critical situations. You'll have them look like a bunch of idiots if they ever arrive too late to help and they were going at a fraction of their possible speed, and the Enterprise arriving too late to help, or being in danger of not arriving in time, happens all the time. So you can't abide by the speed limit in essentially any episode that requires a "ticking clock" element without introducing a plot hole (or inviting a Prime Directive-style debate about how many people have to be in danger of suffocating on a damaged transport before their lives count more than the right of future generations to have pristine subspace, which might be interesting once, but certainly not for twenty-some episodes a season).
So if that's the case, and you keep lampshading that they've gotten a waiver to go at warp 9 because it's a real emergency, then you make the whole thing into another Worf Effect. If every episode requires exceeding the limit, exceeding the limit stops seeming like a big deal, and becomes just a bunch of narrative cruft that you've saddled yourself with constantly bringing up.