Customer to a hatmaker.... "Can you shape my new hat?" - Hatmaker: "Of course ! Costs 50$ additionaly !" - Customer: "OK, can I wait for it?" - Hatmaker: "Done" :)
it's not about rabbit or beaver, it's about the dense and the weight of the felt. And this one is obviously pretty light weight and low dense. But only a few of the finest hatmakers in the world do so nowadays. Most standard Fedoras are pretty "well...standard" weight.
That's not fair. I had to steam my Akubra Federation a bit to get it to take shape. Honestly, it's a thick rabbit which is darn near bullet proof, so your method wouldn't work...
Some hats do be like that. My local hatmaker has rabbit fur hats with a very soft crown like in the video. Meanwhile I've got a cheap wool felt hat that's also bulletproof like you said. Just to make the pinch larger and the crown slightly pointier I had to steam the crap out of it multiple times and spend an hour pinching and shaping it.