They do, it’s called the AMA, American Medical Association. Very tight control of who can be a doctor, how many new doctors per year, etc. which ensures salaries stay high. They also fight against other roles like nurses doing too much doctor-like work, helping doctor pay while harming nursing pay.

At least where I live there’s a shortage of doctors so you can’t negotiate with them . There’s a shortage of nurses too, but if you lose 1 nurse because they’re striking it’s bad but things don’t collapse. With 1 fewer doctor who also knows and is allowed to do things nurse don’t / can’t things get really bad. They also have access to power, are often elected, and are respected.

Nurses need to unionize I think and strike to get something .