Thing is, Voyager managed to deal with the usual AQ suspects whenever it suited them, so the Romulans, Cardassians, Ferengi, Humans, and quite a few Klingons were in the show - arguably only a smidge less than we'd have expected in TNG (given that whenever AQ races were featured, it was a big deal and they just didn't happen to be in the background).
Functionaly, Voyager WAS a continuation of the TNG paradigm and concept, whose key plotting differences adding in various adversaries who by nature would not be featured for long in a show designed to leave them behind. I think that it would not have been markedly different if they'd just KEPT them in the AQ overall, perhaps as a deeper-space mission as has been suggested here, and allowing both VOY, DS9 and even TNG to cross over in a larger picture. The Dominion War was a creation of, and basically contained within DS9, but if they'd had the foreknowledge to plan out something that could involve multiple elements of the TV franchise beyond lip service in an episode or movie here or there (DS9 was name-dropped on Voyager, but NEVER the other way around), we could have had something special.
IMO Stargate SG-1 and Atlantis got that right by not denying the multiple shows were all co-existing in different places of the same universe in their various plots, freely referencing and crossing over plots, characters and backstories.
Mark