One intriguing characteristic of "inertial damping" aboard starships we witness a lot in TOS and the later shows is that even if one part of the ship shakes, not every part needs to.
This makes a weird sort of sense. The damping is supposedly achieved by projecting some sort of a stabilizing field, and we have seen that artificial gravity fields are specific to individual rooms. Just because the field on Main Bridge is getting a punishment doesn't mean that the field on Sickbay would fail to a similar degree and leave McCoy with a floor covered in broken glass, or that the field on Transporter Room Four would let the Genesis Device tip over.
While keeping those on the bridge in place is important, you can make an argument that Sickbay, Engineering, and the Transporter Rooms would take priority. Sickbay, as you explained, is important, since being rocked about might mean the difference between saving a patient and killing them. Engineering, also, is someplace a slight bump could create catastrophic consequences. And, if the transporter room is needed while the ship is being attacked (assuming, somehow, the ship is being fired upon and the shields are down, at least enough to transport), since someone falling off of the pad would be disastrous, as well.
It was a good movie, dont rock the boat.