During my time as a middle manager, I saw how much actual work is needed by management to make this happen. You can't just passively say "take care of yourself :)" and leave it at that. There will always be people who push themselves stupidly like coming in sick or answering work phone calls at funerals. Either due to some imagined pressure to visibly perform, or caring too much about their work.
A manager who truly believes that people need to take care of themselves need to actively prevent people from doing stupid shit.
At one point one of my team members came in clearly half unconscious from cold. I had to urge him to go home and only half jokingly threaten to take his laptop away if I saw him in the office tomorrow.
For some people I had to straight out tell them that if I saw them stepping in to answer support questions while on vacation, there would be stern words. We handle support, and if it is urgent and we need you, we'll reach out.
I think too many managers know this is a real problem, but just passively exploit it.