I think it's weird - but then some people are just really forward. It kind of depends on how she said it - like was it a demand/expectation or a request/letting you know that she would love to be there if you wanted. The first is worrisome boundaries-wise; the latter is kind of sudden and strange, but not worrisome.
I would tell her that you feel strange having someone you just met at your birth given what an intimate experience it is. But that you appreciate her support and desire to help/be there. I would let her know about the problems you've had finding someone to watch your daughter (does she know about that already?) and say what you really need is someone to watch her when you go to the hospital. Maybe at home or in the family waiting room. If she's able to do that then it could be the solution you've been looking for. If she's like, "but I wanted to actually be at the birth" then I'd REALLY think it was weird and back away from this person a bit.