Not really. If a ship isn't registered to the Federation it isn't a Federation ship. Even if for some reason that did not seem like a substantive difference, it does matter to the issue at hand, which is ship naming conventions. There have been HMS and USS Enterprises in simultaneous service for decades. To an alien, they can both be colloquially called Earth ships, but they can both be called Enterprise as they are not registered into or part of the same Fleet.
Which again begs the question: Why didn't Starfleet build any more U.S.S. Enterprises between the decommissioning of the NX-01 and the commissioning of the NCC-1701? Even if there was another Earth ship named the Enterprise, that wouldn't necessarily impede Starfleet from having one as well, as you point out.
Yes it doesn't answer the issue of why Starfleet didn't have another ship called Enterprise between 2161 and the 2240's/50's. I think most likely they would have. In the comics there is at least one Enterprise in between them.
It was already a famous ship name long before the NX-01. It only becomes more legendary through the pivotal and historic exploits of Archer's Enterprise. So I think there should be at least 2 ships by that name after Archer and before Pike.