"A solid blow, even from a wooden training sword, should compress the tyre significantly"
No it shouldn't.
"- much more than he did, meaning he wasn't hitting it very hard"
Yes he was and he clearly was.
"- and I trained with guys who could literally compress a tyre of that type with a blow so that the two sides actually touched."
No you haven't. Never happened physically impossible.
"Honestly compels me to admit that those two guys had no wrists:"
Ore real existance.
"I've also watched a professional re-enactor demonstrate this on a training dummy wearing mail, to prove that it was possible with a sharp weapon. He gave it two solid hits, and then I guess his grip slipped, because on the third strike, he sliced his palm right open through a heavy leather gauntlet, and had to be driven to hospital, with his hand wrapped in a blood-soaked towel. So yeah, it's dangerous."
Nope you never saw that, you're lying. When you start by claiming that you saw something that's physically impossible, that's your credibility gone.