In fact, judging from AGT, the federation still has the upper hand over the Klingons as the Enterpise (now refitted to a Galaxy-X class) beats the crap out of the Klingons before heading home in their (now legal) cloak.

That's the two main factors right there. The UFP has the upper hand in technology, science, and general ingenuity. It's been shown over and over again, and the increasingly blood-thirsty Klingons in the AGT timeline would have been up against humans who were reverting back to 20th century mindsets (arguably even more bloodthirsty than Klingons) but with far superior technologies to weaponize and expand upon. The legal cloak on the Galaxy-X Enterprise along with the phaser lance wepaon are prime examples of this. Even in the prime timeline, we see the more desperate humans of the Maquis do things even more horrendous, like manufacture a self-guiding warp torpedo to destroy entire continents or planets. We know that there are banned subspace weapons as well as old projects that could be revisited (various transwarp experiments, Gensis being weaponized, phased-cloak technology, Iconian gateways, etc). The only thing preventing the UFP from making such things is the Khittomer Accords. If those are tossed out, and the Federation is enticed enough to become more warlike, imagine what they could make.

Phase-cloaked Defiant-class gunships with advanced ablative armor shooting off phaser lances and subsapce singularity torpedoes. Self-replicating, cloaked and self-repairing, intelligent minefields to catch unsuspecting cloaked birds of prey that might try to make a run on Federation planets or stations. Warp or trans-warp self-guided weapons that utilize old Genesis technology to wipe out and terraform enemy planets for occupation by Federation forces as they advance.

This, of course, assumes that the UFP is united in such an endeavor. If humanity fel back to in-fighting, I think we'd see a future much more like what the Mirror Universe has, with a divided humanity collapsing in the face of more united alien threats. The Klingons may have in-fighting and the occasional civil war, but they put that aside when outside threats come knocking on the Empire's door.