I know there's a shortage of mental health professionals and even inpatient beds. But this doesn't fix the problem.

The reason a regular doctor can't do that is because they can't just treat you without a reasonable proof of indication and they can't diagnose you without reasonable evidence meaning while patients are still their source of income they have a hard time abusing this because there are regulations in place to prevent this but in psychiatry/psychology pretty much all countries have not yet put such regulations in place for those which makes a group of people really uncomfortable and untrusting.

we would just have no reason to try to treat healthy people.

It's not just about healthy people. There's also misdiagnosing. Theoretically you can deliberately misdiagnose a patient for some form of benefit which is something a regular doctor can't do due to regulations and safety mechanisms in place which don't exist for psychiatry.

People in anti-psychiatry are concerned about what you can legally do and get away with if you wanted to. How many psychiatrists/psychologists abuse this is a different question (which you also can't answer because you're incredibly unlikely to ever catch this). The main concern is that there are little regulations protecting the patients which enables psychiatrists/psychlogists to do all kinds of harmful stuff all within legal limits. Anti-psychiatry partially consists either of people who got abused (because the law does not protect against it) or people being afraid of abuse (because the law does not protect against it).