There is a very simple solution to this. Have a secure area at the stadium where people can pay a quid or two to leave their prams or strollers. Same system as a cloakroom where people are issued a ticket allowing them to collect afterwards. The money from this would pay for an attendant.
Unfortunately in todays society people are encouraged to claim for everything they can get away with, legit or not. Can you imagine if somebody tripped over a stroller and landed on a concrete floor breaking an arm, leg or worse? They would be advised to sue the stadium management for every penny they could get. The claim sharks wouldn't be suing the person with the stroller they would go where the big money is. The counter argument to this is what if someone tripped over a wheelchair? Would the same apply? And if the SMC didn't allow wheelchairs then they would be done for discrimination.
I think the SMC are just protecting their own interests but it's up to them to find an answer for the club and its supporters.