https://www.midnr.com/Publications/p...hootingranges/
Go here, Michigan DNR maintains a list of clubs that have volunteered to have their info published. Go and meet someone at a couple of the listed clubs and they will probably be able to point you to other clubs in the area.
Check with MUCC, Michigan United Conservation Clubs, they also maintain a list of clubs. Their list may be more all inclusive.
And for the OP:
Don't write off the private clubs, I belong to two gun clubs, wouldn't give up membership in either of them for the matches/leagues and friends. Not many free ranges will you be out shooting and have somebody show up with a 416 Rigby, they hand it to you with a couple rounds and tell you to shoot it���������������� not sure you can have any more fun than that with your clothes on! Pop