For me, FWIW, (probably not much) all thee eps seem to be more "flights of fancy", not to be taken too seriously. I enjoy the original "The Trouble with Tribbles" most, as it made the most sense, relatively speaking. TAS' "More Tribbles, More Troubles" was cute, but increasingly silly. Monster tribbles? Definitely a Saturday morning cartoon. For all the fuss about how DS9 was a darker, more serious bad-ass TREK for the '90's, "Trials and Tribble-ations" seemed to be the cheesiest of the tribble-trilogy. (Tribbolgy?)
The idea of a space station being infested with cute little furrballs the purr seems like an early attempt to give TOS some merchandising legs.
I remember seeing the TOS-R version of "The Trouble with Tribbles" several years ago when TOS-R was being exhibited on broadcast TV. It was kinda neat to actually see the Klingon starship circling K-7 opposite the Enterprise.
Trekmovie did a great job covering the remastered ep, even including a VFX reel.
"Trouble" always gave me mixed feelings, as far back as I can remember. It was cute and funny, but the Klingons and the Tribbles came off as way too cheesy. (Even though Campbell's Koloth and Pataki's Korax were thoroughly entertaining.) I suppose TOS missed a great opportunity by not making K-7 (or one of its sister-stations) into a "home port" the Enterprise could visit.The station seemed to briefly offer a whole new dimension to the show, even moreso than a visit to Starbase 11.
As far as the OP's question about TOS-R somehow matching up with DS9's revisionism, we do get to see Koloth's starship. We also get to see great shots of both K-7 and the Enterprise, both vastly superior to what was shown in TOS. Naturally TOS-R didn't muddy the waters by showing Sisko's Defiant de-cloaking or anything like that. One thing that always perplexed me about the K-7 station was the seeming contradictions in size. One the one hand, it would have to be pretty big; several times the size of a starship to handle people from multiple ships, offer shopping facilities, and to generally serve as a kind of civilian spaceport. OTOH, the images seem to suggest K-7 isn't that much larger than the Enterprise. This seems pretty smallish for what's vaguely suggested. (Well, maybe the images aren't
that conclusive. There may be
some wiggle room.)