I'm also American and had my DS in London. I'm not going to fault you for your opinions, while I know the NHS is great for many people, I did not have a very good experience myself. So I think it's understandable not to want to risk a repeat experience.
But I think you should stay and go private in the UK. You only need to go private for the delivery, after that everything is covered.
In the US you will be liable for everything, and I personally would not put a lot of stock in this reactivated blue cross policy -- I mean, really? That sounds too good to be true. You are pregnant, living abroad, and suddenly they will cover you? And they are not airtight, they are free to challenge paying out for everything and you may find yourself in an almighty struggle afterwards. (My best friend had blue cross in another state when she had thyroid cancer, and they tried to deny so many of her claims, she had boxes and boxes of paperwork from arguing with them.)
God forbid you have complications, your bill can easily go up by tens of thousands. You say you won't have to pay out of pocket with this insurance, what are the deductibles and upper limits? Not to mention, the cost of flights to the US and the complications from being abroad for a few months.
What about your DP? Would he be there too? Can he take off work? Would he miss out on that early bonding time?
Basically, I totally understand why you want to do this, but I think it's a pretty dramatic step to take that in the end may not save you that much money.