For a ship far larger than the original Enterprise and with a far bigger crew, you'd think that sickbay would comprise an entire deck (I'm envisioning their own hospital with multiple doctors on hand instead of just Dr. Crusher, but Dr. Crusher being the head doctor). I mean, 1000 people to care for is a lot!
Budget constraints are part of the reason.
Sickbays on ships should definitely be larger (or at least there should be several) and certainly have more than 1 doctor for catastrophic situations.
The Enterprise-D did have several doctors and multiple nurses (its just the emphasis was on Crusher and Pulaski).
But also, putting sickbay on just 1 deck seems inefficient. In a dangerous situation with casualties, people would need to be brought to sickbay... which would be a problem in case the transporters aren't working (and this can be a problem the larger the ship is).
However, I suspect that nurses (and various crewmembers) would also be capable of performing numerous repairs (and possibly surgeries) etc. which would minimize the doctors involvement and free them up for other patients who need more complex care and in-depth knowledge. Some repairs can probably be done in emergencies by non medical staff as long as they follow directions to use them.
In the 24th century, nurses are more like doctors of today, but actual doctors in the 24th century have more in-depth knowledge for complex tasks.
Kes on Voyager was technically a nurse, but she could also perform various medical surgeries (she certainly seemed to have more free range than nurses of today).
But you can also justify the smaller size of sickbays thanks to technology and automation.
Since various (or possibly most) injuries can be treated and repaired on the go with medical kits which are on every deck, it wouldn't be a stretch to think that critical cases would be smaller number-wise (to the point where
usually they wouldn't overrun sickbay).