I don't think that's accurate. But that isn't important really.Yes they do. But MLB can and did ban Pete Rose from being considered for the Hall of Fame. So, the writers don't get to vote on just anyone they want. There is a screened qualified list. Players are also removed from that list for not having played long enough or garnering enough votes in a prior HOF vote.
The Baseball Writers of America just honor MLB's banned list. From what I understand, MLB does not give a list of names for recognition, as far as I have read.