You have a stressed out mum, with two very young kids. I assume that she must have thought she could get the buggy's on. Then someone jumps up and says nope they are going in the hold or you are not flying... There is the prospect of the miles of walking at the other end, and not enough hands... The lady is going to have a problem....
How about empathy. Understand her problem as it is a problem.
I'm very sorry paying passenger. We have to put the buggy in the hold. I will make sure that we have someone meet the airplane to give you a hand to get the kids to the baggage hall, or we will make sure the buggies go on last and pass a message to get them delivered to the jetway.. (in fact this happens by default at some airports).
To allow buggies to be taken to the aircraft, but to insist they be picked up from the baggage hall, does miss the point!