It should be noted that a lot can happen in twenty years which was the time gap of Yesterday's Enterprise. Like as you mention, a full civil war, contact with the Borg among other things. 20 years earlier, the Klingons and Federation WERE on a more level playing field, enough so that a surprise attack could have seized much Federation territory before Starfleet could fully mobilize. From then it's just a matter of them leaning on the Federation until they collapse. They were fighting near the Archer system so the front lines were quite close to Earth. The Federation considerably buffed up post Wolf 359 when they seemed to start taking their military seriously.
As for the All Good Things bit? I can't say I take that very seriously given it was an illusionary timeline. Too much unexplained to just write it off to brute Klingon strength. The Hobus Supernova could have happened there too, the Borg could have attack them, the Federation/Klingon alliance could have taken them down and things soured when the Klingons occupied then, etc.
The balance of power in the TNG era wasn't so radical that either side would attack the other. It's not like there's no love lost between the Romulans and Klingons so if one side or the other had a decisive advantage, especially the Klingons since the Federation would probably stay neutral, they'd attack.