In the case of the babysitter, they were away from home all day and never checked up. Thus, no yotzei venichnas, and she can be assumed to be treifing up the house.
I originally assumed like you that based on rov keilim I would be allowed to eat in such a house if extended an invite. But my rav advised me against it. Certainly nobody can have a goyishe babysitter lechatchila without checking up on her.