Indeed. And probably for good reason, as we saw in TNG "Unification" that a Klingon cloak allowed our heroes to penetrate to the very heart of the Romulan Star Empire and to linger there for several days unobserved!
The jury is still out on who out of the usual suspects invented invisibility first, and when. Sure, it seems to come as news to our TOS heroes when the Romulans are first not-seen using it, but subsequently it sounds like invisibility is mundane and commonplace - and then comes ENT and shows it mundane and commonplace a century before the "introductory" TOS episode, in the hands of a variety of aliens and
subsequently falling in the hands of both the Earthlings and the Klingons.
Really, TOS "Balance of Terror" is the one hiccup in otherwise acceptable continuity, the piece that never fit the puzzle. Ships that appear out of nowhere and then disappear again are not
news to our heroes in that episode - they witnessed such a thing happening back in "Charlie X" already.
Certainly Romulans and Klingons go neck to neck in cloaking prowess in the TNG era adventures, so it would stand to reason that they would hesitate to have their ships maintained at a base frequented by the opposition... But possibly neither the Klingons nor the Romulans had any choice, as the former had lost their forward bases to the Dominion in the early stages of the war already, and the latter never appeared to have had any.
Sure, TNG "Birthright" suggests that Romulan space is not too far from DS9, but we never hear any exact figures. And there's nothing explicit to indicate that Romulus and Bajor/Cardassia would be border neighbors or anything. Indeed, when the Dominion starts using Romulan space for attacks against the Federation-Klingon alliance, this is considered something of a "backstabbing" maneuver, and not just figuratively...