The question I am trying to answer is why?
I don't think the 'why' could be made any clearer.
The childminder is very likely to have tried other strategies to get him to want to cooperate and to find out if there was an underlying fear or desire that he was trying to communicate to her. If you don't think she did this, perhaps you should ask her why, as it's normal practice to do so within the restrictions of meeting the needs of other children in her care.
If all those other strategies have failed and it is clear to her that the child is determined to stand his ground she has three options.
Take the shoes off him herself. You don't want her to do that.
Allow him to come in with his shoes on. Moves the boundary he is feeling for, causing him confusion and totally undermines the authority that enables her to keep him safe and secure in her care and manage his behaviour appropriately. It also means that the problem is still there the next day.
Let him stay in the hall until he is ready to cooperate. You don't want her to do that.
So what do you want her to do?
Children of this age go head to head to try to get their own way. That's why it's called the terrible twos. They are trying out the boundaries and working out how the rules work. If they come across firm, fair, clear boundaries, they work them out quickly and become happy and secure within them.
Whether you like the terminology or not, your DS needed to learn that he can't win all of the battles. Maybe he stood his ground for longer because his previous experience was that he would win in the end. Only you know if that's the case.
If you don't want a childminder who will give your child firm, clear, age appropriate boundaries perhaps you need to look for a different one. In order for your DS to thrive in her setting he needs to know that you support her methods.
You pick you battles with two year olds. You look for other strategies to avoid conflict. However, conflict is sometimes unavoidable and, in the real world, the adult has to be in charge in order for the children to be safe.