The domestication of dogs happened a lot longer ago, and probably a lot more organically (more like what you're thinking). The wolves would follow us around while we hunted (or, alternatively, we'd happen on one of their meals and chase them off), and the less human averse among them got more food.
Cats happened about the same time agriculture did. They were attracted to the rodents, and less rodents meant we were happier - and they were happier with a reliable food source. At the beginning it might have helped to not be afraid of humans, but we probably didn't mess with them too much because they were helping us out with all the mice in our grains.
So the difference in behavior is clear, not just from a social species standpoint, but from the domestication path:
The European Wolf was domesticated to dogs as a hunting partner; the African Wildcat was domesticated to kill the rodents that devoured our foodstuffs. For the few few thousand years that was probably enough - they didn't have to make friends with us, just keep our food safe.
Earliest cat domestication evidence is 9,500 years old; earliest dog domestication evidence is about 36,000 years old.
They've still got quite a ways to go ;) Now if we could just start breeding them for social/intelligence instead of how weird and fluffy we can make them. Hard to morally advocate for breeding when the shelter/feral population is so damn high, though. (I feel the same way about dogs, though, tbh).