Is there any evidence, anything whatsoever, that they actually are so powerful?

Maybe. Maybe not.

His point is that Anthropic is likely doing this on purpose for their own IPO and to counter the other IPOs.

They didn't say they are. They said it is PR which is a type of marketing which has to do with perception not reality.

Are you asking somebody what evidence they have that their observations are wrong? Like “I see you have an opinion there. What facts are you aware of that disprove it?”