Yikes, third nanny this year is indeed a huge red flag.

But, that also may work in your favor since at this point they have to be seeing that they are probably part of the problem. The fact that you were able to have a direct conversation with them, and that their response was to say that they wanted you to stay and they are willing to try to work on it, is encouraging.

If I was on my third nanny in a year I think I would try awfully hard to make it work.

Good luck.