Parliamentary systems typically choose PMs or whatever from elected representatives, but it’s true enough it’s not strictly required, and he certainly had the support of the Nazi base.
The other important lesson to learn from the example, other than the Nazis never won a fair national election but seized power anyways, is that Hitler could not run for office because of the crimes he was convicted of. You know, like trying to overthrow the government.
Disbarring a fascist from office doesn’t work if you let him out of prison with a slap on the wrist.