I feel like this is similar to asking your insurance to cover NyQuil and Tylenol. Not exactly unreasonable for them to stick to what they’ve been covering so far, and not expanding into otc just because it is birth control. It’s equivalent also to asking them to buy your condoms.
The thing is your doctor can absolutely prescribe you Tylenol (or at least acetaminophen), and your insurance will cover it. I've been prescribed ibuprofen before. Just because it's over the counter doesn't mean your doctor can't prescribe it, and the OTC things insurance will cover already is pretty broad: diabetes supplies and COVID tests, for example.
I would've assumed the insurance companies would weigh the cost of birth control vs. the costs associated with pregnancy and childbirth and that they'd prefer the cheaper option, but I'm not sure how exactly medical insurance works in the U.S so maybe I just have an idealistic view!
I was surprised there were no incentives from my pet insurance for me getting my dogs spayed, but they were quite clear that they didn't cover any costs relating to having puppies and I would hope medical insurance would cover pregnancy and childbirth...