The toxic culture is coming from the outside. They have no reason to trust you, because dozens of corporations institute wellness programs and talk about caring while still punishing people who take time off. Plus, some of them have just internalized the idea that they should just work through it.
One thing I have heard can help is if the boss takes time off, so since you have some authority, maybe make a point of taking your own time off and telling them when you do?
Too Healthy (leminal.space)