If the club has nothing else the grass should be cut every week regardless if someone thinks it should or shouldn't be.
When the season starts it seems folks are gung-ho to do the job..........for the first few times and then the novelty quickly wears off and it ends up being one person doing it ,or it goes undone . Trying to put a volunteer schedule together usually will not work.
There really aren't many choices but here are a few:
*The president, vice-president (and possibly a club member)are responsible to cut and trim the grass. Need motivated ,dedicated individuals in these positions,
*Mandatory club participation. Everyone is responsible to take a weekly turn. If someone can't physically do it then they get another club member to fill in for them. If there are enough members each person may only have to do it one time. Small price to pay to keep dues low. Install a combination lock on the building so everyone has access to the mower and a club officer is responsible to keep the gas cans filled.
*Raise the dues and hire someone to cut the grass .
I've been at clubs where all of the above have worked well.
If your club is fortunate to have some folks either officers or members who cut the grass every week without question or complaint then the members should thank them and do everything they can to keep them happy.