I think the problem really came down to the size of the fleet, not the number of ship variants. If each fleet was 20-30 ships instead of 100+, it would have been way more plausible overall and not really taken anything away from the scene. It was a mistake, but apparently someone really wanted to hit control V a bunch of times.
Why?
The Federation is comprised of over 150 alien species in late 24th century at the time of First Contact. More if you count Insurrection and the fact they admitted new members.
Each species solar system has its own shipyards... and if they are a fresh species (just achieved Warp travel), and admitted into the Federation, it stands to reason that Starfleet would bring those solar systems up to capacity in line with other solar systems ship-building capabilities).
SOL alone has what... hundreds or possibly thousands of drydocks (each capable of building a starship).
There are shipbuilding facilities in Earth's orbit, in orbit of Mars, in other parts of SOL... and the situation is likely the same in each/every member species native solar system (such as Vulcan, Andoria, Tellar, etc.).
With only 100 drydocks per each member species NATIVE solar system (and there's bound to be MUCH more than that), you can easily allocate 10 drydocks per solar system to build a given ship design.
And in a short span of time (a couple of weeks or less - thanks to transporters, replicators, tractor beams and automation), you'd end up with AT LEAST 1500 ships (if 150 systems were building ships) of a given design.
There were also 7000 active ships in the 23rd century Starfleet. That's with much less member planets/species.
By 2371, we've seen Voyager had a hull registry of 74656. It stands to reason this would indicate a number of ships in the fleet... or over 10x more ships than what SF had over 100 years earlier.
Its perfectly plausible.
Also if it takes a while to travel at Warp, you'd need more ships for patrol of existing Federation space, exploration, etc.
So, having 100+ ships of a given design would basically be 10% of a total number of ships of that class.
Not a big deal for an organization as big and advanced as the Federation.