I may be an incredibly naive European, but when you’re burned out, can’t you just get sick leave from your employer?
clearly an option for OP. why do you think its otherwise?
Secondly, in case of burnout (or similar), it’s not unheard of (at least over here in .nl) that you stay at home for extended periods of time, and gradually restart picking up your workload. That kind of thing is very different from vacation.
You should never have to reveal the nature of any health conditions to an employer to take sick leave.
i've had aquantances that walked out with 3 days of sick leave (in Germany) - not even connecting to a weekend