For everyone who thinks that the solution to these on-board instances is for GAs to better police regulations, the difficulty is this. Each layer in this customer service chain typically either wants to be the "good guy," or doesn't want to deal with problems. Therefore, rather than the GA refusing to allow the stroller down the jetway (as might be policy), they say something like, I don't think there is space on the aircraft for that, but you can go on board, take a look and gate-check it if there is none. By doing, that, they force the FA's to play the "bad guy," and avoid having to deal with the difficulty and conflict themselves.
Note, too, I have seen GAs give pax gate-check tags for oversize bags or strollers, and allow the pax to carry them down the jetway with the admonition that the pax should leave them at the end of the jetway to be collected for hold stowage. And I have seen many pax ignore this admonition and carry the bag that was supposed to be gate-checked onto the aircraft in hopes that they will be able to find adequate space.