Trifle by your reasoning, any student should do PE and if they can't do PE they shouldn't be in school.
If a parent sends a letter in to excuse their child from an activity, for whatever reason, then the parent should have reasonable expectation that the request in the note is honored. If the teacher has an issue with the parent's request, they need to take that up with the parent, not force the student to participate regardless, potentially doing harm to said child in the process.
It's no different from when I was at school and I broke my (very strong, complex px) glasses and it would be a couple of days before they could get me my new ones. My mother wrote a note asking that all my teachers allow me to sit front and centre in the classroom so I could see the board (regardless of where I usually sat in their class seating plan); that I be excused from PE full stop, as without my glasses I get dizzy and have no real depth perception; and if at any time during the day I started to suffer from headaches, I should be excused to the office to call home. The school had no problem making the adjustments for those few days and once my new glasses arrived everything went back to normal proceedings.
By your reckoning, I should have spent those few days at home, or got a note from my doctor, since I couldn't fully participate in every class ... despite being perfectly fit and well in every other way than I couldn't see properly. If I'd have suggested that both my parents and teachers would have laughed me all the way back to my classroom!
Being fit for school does not mean you are fit to participate in every facet of the school day and as a fellow teacher I can't understand how any teacher would have that expectation.