I just think it's a poor fit into the Trek universe as we now know it. Not only is there no other mention of Kzinti, but none of Slavers or stasis boxes. It stands to reason that they would've left artifacts all over the Trek universe and their impact would be felt, as is the case in Known Space. Sure, there are plenty of ancient races we only ever hear about once, but the Slavers were supposed to be this vast galaxy-spanning power, and it's stated outright in the episode that their boxes are highly important, sought-after treasures. If they were part of the Trek universe, it's hard to believe they would've gone so totally unmentioned outside this episode. That makes it hard enough to reconcile even with what came before TAS, let alone everything that's come after. The story just works better in Known Space, because it was meant to happen there and was an outgrowth of ideas already established there (in World of Ptavvs and "A Relic of the Empire"). It's a much more awkward fit in the context of Star Trek.
You make good points about the Slavers. Though I do have a few thoughts that could explain them not being mentioned. For one thing, I seriously doubt humans at the time of Enterprise
would have known anything about the Slavers anyway due to the Vulcans beyond perhaps a few boomers who may have heard a tale or two tales and they may have only considered the Slavers to be alien myths.
Another thing to consider is that as time passed and civilizations advanced, the Slaver stasis boxes would render fewer advances and thus be less significant. In fact, considering the Slaver weapon only had one setting that was more advanced than the technology of the time period when it was found, it could be that the stasis boxes were already becoming less significant or this possibly was the beginning of them becoming less significant. Perhaps only a less technologically advanced species like the Kzinti would find much in a stasis box to be of any practical use. Perhaps more technologically advanced species, if they happen to seek out the stasis boxes at all, do it mostly out of curiosity more than out of trying to find more advanced tech and thusly the Slavers aren't much talked about outside of academic circles.