Donations are not entirely without strings. In theory (and usually in practice) a charity has to work towards its charitable goals; if you donate to the local animal shelter whose charitable goal is to look after dogs, they have to spend your donation on things like dog food and vet costs.

Charities have reasonably broad latitude though (a non-profit college can operate a football team and pay the coach $$$$$) and if you're nervous about donating you can always turn a lump sum donation into a 10%-per-year-for-10-years donation if you feel closer monitoring is needed.

Unless the donors were already owners.

Officially, yes, but the whole situation with Altman's firing and rehiring showed that the donors can exert quite a bit of control if their interests are threatened.