It's not a case of the manager not supporting her, it's a case of the manager putting something that could have been informal - "I'm happy with your performance and if you need to take some breaks during the workday I support it" - and made it a formal thing that is risking getting her fired.

The manager in question has admitted they fucked up and didn't realize how much HR would try to force my friend out for being a problem.

I don't agree. There's nothing 'informal' about someone under you telling you they're suffering from burnout. It's not a water cooler topic of conversation. Legally, going to HR was the correct thing to do.