I think if I had been the babysitter in that situation, I would try redirecting. The child sings Happy Birthday and I may ask her to sing another song, say the ABC's or The Wheels on the Bus. Not sure if she has actually tried this approach since I was never told about this directly. Since I promised my wife I would not get involved, I will not bother to mention it to the babysitter directly either.
Happy: Yes this is a phase, just as you say.
maisha: The babysitter needs to realize that there are other kinds of people out there and if she is going to take in non-witness kids, she needs to realize that these things can happen. If short, grow up, the whole world is much bigger than they are.
blondie: This babysitter is by far the cheapest considering how good she is with kids. A daycare center would easily run us $200+ / week for children her age. Since my wife and I had agreed a while back that she will go to a center at age 3, I am not too concerned. That decision is based on the fact that a center will engage the children more than a sitter. That said, she is still learning quite a bit thanks to the sitter. I will say that on balance, she is a good sitter and will certainly recommend her to other, in spite of her little quirks.
Qcmbr: I will have to say on balance, she is a good sitter and worth her weight in gold. But I would recommend pulling the child at age 3 though to better prepare the child for kindergarten. Our oldest was in a center since 4 months and you can see a real difference academically.
jgnat: For real, what can I possibly say to my youngest for singing Happy Birthday.
WTWizard: I do think about the molestation thing but she will be leaving soon. I don't think this babysitter is prone to molesting the kids, so no worries. Will keep the kids involved in holiday and birthday celebrating though.
Billy: I love it, LOL. Got to try it.