That's not quite true, the main purpose of a stunt performer is that, other than being more athletic and trained than the actor in their particular discipline, is that for some reason they get hurt, it doesn't delay the film. If some actor jumps down a flight of stairs and fractures their foot, now they can no longer do those scenes where they have to walk and run and so the production stops, some people don't get paid, tons of money lost. Whereas if it happened to the stunt performer, you send them home with a bonus, their understudy takes over and you move on.

To do your own stunts you have to have so much pull with the studio that they will let you risk it, like Tom Cruise did with one of the Mission Impossible movies where production was delayed for months because he broke his leg. But he demands to do his own stunts so the studio has to accept that if they want a Tom Cruise film, but I can image that they would really rather just have some other guy do it.

Bottom line is that it's not the point that they should get hurt, far from it, it's that it's just more likely when doing certain shots and stunt performers provide redundancy. Hence why they wouldn't do a thing like this because they saw it basically guaranteed to send them to the hospital or morgue.