What a terribly painful situation. I think you are correct. Your daughter-in-law liked having the free daycare but she didn’t like the fact that you had a loving relationship with her son. How sad that she feels this is a competition. Children get different things from parents and grandparents. They benefit from having both in their lives. A loving relationship with one in no way diminishes the relationship with the other. In fact, the more we fill children up with love, the more loving they become.
I am concerned that you felt you needed to be constantly apologizing and explaining. From your description, it doesn’t sound like you had anything to apologize for. She really should have been grateful for all the help and support you were giving her. But — you can’t be the one pointing that out. She’s only likely to get defensive about it.
Sadly, there is little you can do to change her mind. If your daughter-in-law is so insecure and jealous, it is unlikely that you can have a rational discussion about it with her. All you can do is keep offering to give the young couple time off by taking their son for a weekend here and there or even for a longer vacation. If she feels she is getting something she wants, she will probably respond positively.
I sincerely hope your son will at some point be able to reassure his wife and help her understand that they are getting far more than only babysitting from your involvement. Being part of an extended family, when it is functional, is one of the best safeguards for life’s inevitable disappointments and tragedies. Love is what gets us all through.
I wish you well.