IDK about the doctors, but many rabbanim participated in them.
Not everything could and should be in a signed public letter.
I specifically picked up on the claim to backing of not only local Rabbonim, but also doctors.
You're right that they have to be careful about what they put out publicly, but when I made a Bar Mitzvah this year, I called up a prominent Lakewood doctor who's been mentioned several times in this thread and asked him for safety guidelines for the event. His response was, "Don't do anything remotely like the hefkerus tha's going on here (in Lakewood), and you should hopefully be fine."