It is a mess, to the point where I have a hard time deciphering what the author was even going for. But I guess I get the gist.

You're definitely not going to be able to make this as a paladin subclass. Paladins don't have enough design space in their subclass to squeeze in an animal companion. So it'll have to be its own class.

I suggest you copy most of the revised ranger's animal companion feature, and offer different types of dragons instead of the animals normally offered. They'll need custom stats, because the dragons in the MM vary too strongly in power. For the rider's own abilities, I'd suggest you keep those limited to leave more space for the dragon. I'd also tie them to the subclass, so that there can be some variety among dragon riders aside from the type of dragon they ride. Perhaps one martial, one divine and one arcane subclass.

Back in 2016 (shortly before the revised ranger came out) I made an attempt at a beastmaster class. It never got finished, but it illustrates my suggestion. It's basically a ranger with a buffed-up animal companion, and to accommodate the stronger companion I've split up the abilities of the character. The Huntsman fights like a ranger but doesn't cast and the Shaman casts like a ranger but doesn't fight. I would've added an arcane class as well if I'd finished it. I think something like this would be the way to go if you want a proper dragon rider class.