At the end of the day, the Typhon Pact isn't built on what makes the Federation strong. They're all individual empires of opportunists. Their cooperation will be short-lived; a few years, perhaps a decades worth of story time.
You're forgetting Enterprise
. It showed us that in the 2150s, the Vulcans and Andorians were both imperialist powers constantly on the brink of war with each other, and the Tellarites weren't much different. And yet within a few years, they learned to move past their differences and forge an alliance that would last for centuries. Societies can change and grow. The factions in power can be thrown out and replaced with leaders who have different agendas and goals, as happened with Vulcan; or their leaders can just learn to make peace with former enemies, as with the Andorians and Tellarites.
Eventually, one race will depart and not long after it'll splinter. Hate is not a strong enough binding force, thankfully.
But they're not bound by hate. They're bound by mutual self-interest. In Destiny
, President Bacco convinced them that they needed to band together against the threat of the Borg, that they'd be stronger together than apart. But she used heavy-handed, borderline coercive tactics to ensure their cooperation. So the Pact's founders realized she was right -- that they would be stronger and better able to defend their self-interest if they cooperated for their mutual defense. And yet they didn't appreciate the Federation's 800-pound gorilla act, seeing it as just another imperialist power throwing its weight around, and so decided to form an alternative union so that neither the Federation nor anyone else could push them around anymore. That's not hate, it's just the desire for autonomy and self-determination.
So it's a mistake to call them a bunch of "evil empires." Most of them see the Federation -- or their Klingon allies, in the Kinshaya's view -- as the evil empire, the powerful cultural imperialists throwing their weight around and threatening other nations' sovereignty. They see the Pact as a defense against such domination.