Did modern alligators and caimans come from crocodiles

I know that crocodilians have been here since the time of the dinosaurs. I also know that crocodiles came first during the Cretaceous. But alligators have been here for a good 37 million years. Did alligators come from crocodiles when the crocs swam into freshwater areas?

I think it was like this:

It started off with crocs in saltwater. The crocs swam into less salty areas. In Indonesia they evolved into freshwater crocodiles and stayed that way. In the USA though it was different. Some crocs at the tip of Florida lead to the modern american crocodile. Some though evolved into freshwater crocodiles. But it didn't stay that way. With the abundance of large prey the ones with wider snouts were selected. These wide snouted crocodilians still weren't alligators yet though. They were still crocs. These wide snouted crocs continued up north where they were selected once again to have the most cold resistant ones evolve. These cold resistant crocodilians eventually evolved the hole digging behaviour for surviving the dry season and having the eyes and nose above the ice during freezing winters, almost being in a hibernation state. They were now alligators. But they still weren't modern alligators. A few more characteristics like the use of tools evolved and the modern american alligator was finally there 2.5 mya.

This is just my hypothesis on how alligators evolved much later than crocodiles.

And since the alligators and caimans are 2 groups in alligatoridae did they have a common ancestor?

I do beleive so. I believe it was prehistoric crocodiles that gave rise to modern alligators and caimans.

And also if alligators came from crocodiles(Which might or might not be true) Why isn't it more like this:

Crocodylidae -> Alligatoridae -> Alligatorinae -> Alligator

for alligators and

Crocodylidae -> Crocodylus

for crocodiles?

I mean that makes sense that alligators evolved from crocodiles and thus should be in crocodylidae which is the same group that has crocs.