On the other hand, I do get that the premise is supposed to be 3 factions duking it out. The Sc1 setup was perfect: The Zerg had reason to attack Terrans and Protoss all because it fed into their innate drive to consume and better themselves. The Terran had reason to be hostile to Zerg (undestandably) and the Protoss for their careless and seemingly inhumane treatment of their worlds/people (and because humans are always more naturally inclined toward xenophobia generally). The Protoss had reason to be hostile to Zerg (once again for oobvious reasons) and the Terran for their belligerent attitudes and flagrant disregard to their planets, themselves and aliens as well as their role in providing a potential pivot point advantage to the Zerg.
Course, the shift in focus to specific "important" characters in later installment meant that it made Terran and Protoss hostilities less "justifiable" (or naturalistic) - as borne out in the hackneyed matchups we get in WoL. The Raynor vs Selendis fight needs a huge amount of willing suspension of disbelief and the Tal'Darim are presented seemingly in a manner only just to provide Protoss enemies to fight even if they have been represented elsewhere. The Shadow Corps in LotV has the whiff of Tal'darim on them as a result now, too. They're there just to provide a TvP matchup because there seems to be an inherent
need for it even though the lore/story kind of makes it hard now to make it seem naturally occurring without gimmick.
I've always had mixed feelings with the UED but I don't agree they had no role in continuing the 3-way-fight you expect of Sc. I know this is debatable, but at the time of Sc1, the K-Sector Terrans were essentially intimated in the story as the weaksauce faction of the 3. Even though all factions suffered crushing blows, to me the Terrans always felt behind the 8-ball in any matchup with the other no matter how weak the other two were. The UED, as a concept, was the opportunity to give the Terrans a shot-in-the-arm in terms of power levels and some potential for antagonism (as you've noted in the past, you've felt Mengsk shouldn't be agressor against the Protoss or Zerg when he gained power). Sure, it was a little contrived and convenient but the UED potentially gave us a reason for us to sustain the core concept of the 3-way conflict that is supposed to be what Sc is about. The way it turned out was quite a bit of a hit-and-miss however (that they focused on Mengsk for the majority is justifiably disappointing and then just made to disappear in the end) but if nothing else, the UED have my kudos for having the potential of making the Terrans seem more "relevent" in the 3-way conflict.