Johnny made him what he is.

...in Cobra Kai. He was in Eagle Fang for like 5 minutes!

I think he suits Cobra Kai better his style of fighting fits in better there !

He's better as Johnny's student in Cobra Kai and Eagle Fang. I adore him as Eli in Miyagi-Do, he's cute and handsome with his short hair. He got his confidence back and reunited back with Moon and became champion under Miyagi-Do. But I still love his mohawk which made him a badass.

No he wasn’t technically in Eagle Fang long. BUT Johnny was with him as long or longer than anybody else. For Hawk’s personality…I think Johnny is his best choice for Sensei. And since Johnny has Eagle Fang now…
That makes the most sense I think. Hawk’ excelled in Johnny’s version of Cobra Kai…call it Pre-Eagle Fang, Cobra Kai-Lite whatever…. Kreese and Lawrence’s version of Cobra Kai as you know, proved to be drastically different.

EDIT: Something to think about…now that Kreese-less Cobra Kai is going
to become yet another version of Cobra Kai under Silver. It will be even more ruthless, violent, and psycho…and now that Hawk is half-way in a good mental place I don’t particularly want to see him go running off to Silver…do you? I mean that’s Cobra Kai now…at least in name.

^Yeah I see where you are coming from.I do think Johnny is the better Sensei for him but I always picture him in the Cobra Kai outfit it seems to bring out the fire in him.I hope Johnny will go back to Cobra Kai at some piont and take over from Terry silver and maybe take Hawk with him.