Most ships like that might not have a "bridge" as oppose to a control room, or pilots chair.
Also in the two instances where the numbers for Enterprises come up, firstly the Computer seems rather clever. Assuming there was a non-Federation ship, or not Starfleet ship named Enterprise, it was likely no longer around by the time the NCC-1701 was commissioned. The computer was clever enough to pick up on Scotty wanting to see a Federation ship named Enterprise, without going to said one of the aircraft carriers named Enterprise. So we can assume the computer is clever enough to exclude things that don't logically match the questions from a given person...to a degree,
The other instance was Temporal Investigations. They would of course assume it was a Federation ship named Enterprise since Sisko saw it on Defiant's viewscreen and he had not mentioned being in a ocean or planet of any kind. However the NX-01 was not considered at all (mindful of course that that series would not exist for several more years at that point). Perhaps they could tell it would be a Federation ship just by how Sisko said it. Though we did not actually hear that part of the conversion. We got the flashback of it happening, then after the break the investigators questioning which Enterprise it was, saying there had been 5...6. The NX-01 is not on their minds, nor is the ring ship Enterprise. So we can assume there are some logical bounds for what is usually considered when one wants to see the Enterprise or when one talks about an Enterprise. But there are always possibilities...as well as things not said. Temporal Investigations missed at least one spaceship, more than likely two that were named Enterprise, probably because stuff happens to starships named Enterprise, but no one talks about Archer's ship like they talk about ships with a registry of NCC-1701.
Thus, an opening. Not for anything grand, or even anything important. Just an opening for something named Enterprise between 2161 and 2244. There isn't much in the away of opening after that, though again there could be civilian ships or craft that use the name Enterprise that would not be considered Federation ships, They would also be more or less unimportant. Like a sail boat in a marina in Richmond named Enterprise or a planetary survey ship named Enterprise registered on Alpha Centauri. No one would pay them any mind in most history text.