So within multiple universe logic there is a theory, that the universe prefers things to remain the same.

So, yes, every movement of every sub atomic quark leads to every other possible outcome of every situation, but if you shove a technological wrench in the machine and shunt yourself sideways in probability a bit, you're going to end up in a universe that's as similar to your own as the universe can manage despite your meddling. This is why, despite the incredible odds, they end up in universes that darkly reflect the "prime" universe as closely as possible.

Theoretically, they could find a universe where Bajor was completely annilhated by cardassian shock troops, or the alternate universe where after the bajorans threw off cardassian oppression they became strongly militaristic and were antagonizing the federation, and they could find a universe where terok nor is populated by slug people and none of them were ever born. Maybe they could even find a universe where instead of humans a race like the sheliak evolved on earth, it's not just crazy luck though, it's because the universe abhores an abnormality that when they shift over they end up in a similar world.

Think of it like a branching tree, you've got your trunk before Kirk's (or Jonathan's) incursions, then you've got main branches, then you've got an infinite number of smaller branches, in most of which you're correct and the later series characters like Picard, and the DS9 bunch don't exist, but we happen to see the one branch where they do.