To me the kind of relationship for want of a better word that OP is describing to the various women who work for her, was quite usual in the 1950s and 60s where although cleansers, home-helps, housekeepers or whatever they were called were paid to do a job, a pretence was kept up in many families that these women were family friends.
My mother had this kind of relationship with our daily help, but today I think most people, both employers and employees , prefer a more professional relationship.
It sounds nice to say, "she is as much a friend as an employee" but it has its disadvantages: it becomes nearly impossible for the one to complain about the way the other is behaving and makes terminating the agreement practically impossible without a lot of hard feelings.
I would never expect an a person I was paying to do a job to become a friend - if they do become a friend, well and good, but it would make it harder for me to ask them to do the job they had originally come into my home to do.
So yes, to me OP, you are being unreasonable.
If you are satisfied with this person's work, keep her on, but don't try to make her a friend.