The thing is, without some sort of barrier, they would be completely swamped by the public trying to get them to record these, and then would be practically forced to stop doing it.
They will be forced to stop when it becomes more popular anyways, unless some sort of lottery systems gets introduced or fees get bumped up into a $1000 range. But even then I think most will pull out of it after a while, especially if it gets popular, hard to imagine it is an exciting activity to do hundreds of such video messages every day.
They could add a "lottery" element to it. A less evil implementation of that would be to ask the performer what size stack they want and just keep a list of the latest n requests (or requests up to m days old). Automate a reply "sorry you weren't able to get a message from $celeb, blah". You'd need heavy filtering to avoid abuse (automated re-requests).
