We could invent all manner of excuses for what in reality was lack of dough and time. For example, perhaps Constitution class ships tend to operate together because they share the same speed (not necessarily the fastest in the fleet, but still the only ones with those exact speed characteristics) and thus work best together. When Kirk and Commodore Wesley have their M-5 fun, another, slightly faster or slower, newer or older type of exploration-oriented heavy cruiser could flock together elsewhere for wargames of their own.
We could also argue that all the deep space ships at that time were either Constitutions or closely related types, though. After all, Spock in "Tholian Web" identifies the Defiant as a "starship by configuration", in a situation where the only sensor input he gets is vague optical imagery. If Starfleet were operating Miranda or Constellation lookalikes at the time, such identification would be less justified.
I vaguely remember that an earlier script version of "Amok Time" would have given some starship names. Much like an earlier version of "Journey to Babel" spoke of another simultaneous attack against another delegation-carrying starship. Just fanfic, or actual fact?
Timo Saloniemi