I can see crew size maybe being a limiting factor for maybe some missions, but for the most part, not really.
Crew size and lab space are indeed the main limiting factors.
The
Galaxy-class had dozens of labs, hundreds of crew and could accommodate up to twenty times it's basic complement (with a typical overage of 25-30% apparently).
The
Intrepid-class had several labs, about 100-150 crew and a maximum limit of about 350-400 (based on the episode
Prophecy).
The
Nova-class appears to have a crew complement of 78-80, maximum limit is unknown but might be anything from 180 to 300 depending on whether it has Defiant-style bunk rooms or regular state rooms.
Also, internally, the ship's design could have changed in the proceeding 26 years to accommodate more crew.
Logically, that's true.
However, the Constitution, which is a similar length to the Intrepid and has less volume overall, has a significant higher implied evacuation capacity.