Is it an ego thing?

Certainly. I think it's much the same thing as when Will Smith refused to do those gay scenes. I know he's said he regretted that later on, but in a lot of markets, that definitely would have destroyed his macho appeal. But his career imploded anyhow, it didn't matter.

They get beaten up, (or rather, take hits) but they have very specific clauses in their contract on HOW they take hits. Here's the article:
https://screenrant.com/rock-vin-diesel-jason-statham-contracts-movie-fights-lose/

For example, you will never (and have never) seen Statham, Vin Diesel, or The Rock take a beating the way Bruce Willis and -- as OP pointed out -- Van Damme and Jackie Chan have taken beatings. Heck, in multiple movies Bruce Willis' characters are beaten within an inch of their life (Last Man Standing), and Jackie Chan will occasionally lose his fights and resort to running, if he needs to. Van Damme has also portrayed characters beaten within an inch of their life (especially Cyborg and Universal Soldier).

It helps build character dynamic when the characters have to reach deep down, just like in many Stallone flicks (although, he's rarely beaten to a pulp like the other characters I mentioned. Very few films show Stallone losing a fist fight, in fact I can't actually think of one outside of Rocky III and maybe Lockdown).

Anyway, it definitely takes away from the appeal of the character if they can't lose or if they have to only lose under very specific and very contrived circumstances (as depicted in the Fast & Furious films for all three of the eponymous headliners in OP's post).

Statham has never been in a movie where he's been bested the way Stallone's Rocky has, and The Rock has never been in a film where he's been steam-rolled in a fight the way Jackie Chan has, and Vin Diesel certainly has never been in a movie where he's been bloodied in a fight the way Arnold was in the original Conan. Although, Vin Diesel did almost showcase some three-dimensional characterization for Richard B. Riddick during the first half hour or so of the last Riddick film, when he was left for dead and had to straggle and scrape his way through some harsh survival circumstances.

Did Arnie have a clause in his contract not to get beaten up though?

I remember an article from back then saying that both of them refused to show themselves "down" or "on the ground". Or something like that. I can't recall the exact details

Him, Vin & Statham honestly think themselves as indestructible bulldozers. Which, I'm gonna be honest is not entertaining to watch.

I've been tired of him for awhile now. It's not that I don't like him, I just want variety. It's bad enough that we don't get variety in our movies, we could use some different faces in the big budget type films.

Same with Jackie Chan. He gets beaten up & humiliated in his films but he's still beloved by millions across the globe.

When the hero is down, all bloodied up & humiliated, it makes his eventual victory that much sweeter. He actually overcomes an obstacle. Instead of being a bulldozing machine.

That's why I couldn't get into Jason Statham & Vin Diesel's action films. Same as Dwayne Johnson but at least The Rock is charming when given the right script.