S.M.
Unfortunately with any management position no one will ever do as good a job as you, and each worker does things differently. Management requires more adjustment on the part of the manager than on the employee- good management anyway. My husband have had a few caretakers over the years and have noticed great care but also tremendous variation in each. One cleaned more than the one that got down on the ground and played, but all were productive in their own ways.
It sounds to me like this woman has a certain style. So long as your son is safe and seems happy, it would probably be better for you to adjust to it than to force her to adjust to a standard you may not be able to define, which means she'll never be able to meet it and the whole thing could go sour. Then you'll have to start w. someone else and who knows what you'll get.
I also know that managed employees need a lot of praise and validation, so be sure to provide that. You want your son's caretaker to feel appreciated.
Finally, I don't think a child is going to get all that much out of an adult that sits on the floor and plays. Maybe in small doses, but what a kid wants is other kids. Fending for himself is probably not at all a bad thing.
Incidentally, our first nanny was by far our best, so it was a tough adjustment following that act, but I guess it was beginner's luck.