So let me get this right - if the child falls ill it's more important that their unqualified parent can get to them than it is for them to already be in the care of a qualified nurse. And this is what is called the "child's best interests"?
It isn't an either/or situation. You are extremely rude about day care workers but (imo) very generous about the abilities of nurses to diagnose/treat illness. Your mother would have to call an ambulance if there were a serious problem, wouldn't she? I assume she wouldn't reach for the scalpel and proceed alone? In the same way, the person in charge of daycare would call an ambulance.
The presence of a parent is to ensure that the child feels safe and less stressed than he would otherwise be. In an emergency, an ambulance would still be required.
I'd like to ask you, though - do you honestly think it's ok to despatch your 2 year old to an unfamiliar person 200 miles away? Or are you so tired and desperate that you aren't thinking straight? If it's the latter, you need to tackle that problem.
I'm a granny of the mother-in-law variety - in order to help out, we moved closer to my grandchildren at the request of the parents. We do a fair bit of child care but my priorities are the same of those of the parents - to do what is best for the children. If I were to disagree with decisions they made, I'd keep schtum. They are not my children, and it's not for me to be huffy or critical of decisions made about their welfare.
I see mine a lot, including whole days while their parents work, and even I wouldn't dream of taking them away for weekends just yet. The elder one is just 3 and very verbal - he does come to stay overnight, but we live a ten-minute drive from his parents.