It is all ridiculous.

It honestly depends on the individual hospital. They almost always have a contract with insurance that they can't accept less, but nobody's checking and it's common for them to do so. Some hospitals are reasonable and especially if they see if you have low income/assets, will settle with you. Others don't care at all and will not accept any less and just send you to collections or sue.

Well of course the hospital billing department is going to tell you that you have to pay the entire bill. What did you expect? Ask to speak to a manager, make a specific counter offer, and threaten to pay $0 if they don't accept. It only takes a few minutes and even if they refuse you won't be any worse off than before. (I agree this is a ridiculous system.)

I've said things like "really? there's nothing you can do about the cost?" but that's a really weak push for it... I just hate kind of being a dick about it (asking for a manager and making a counteroffer makes me feel pretty shitty, but I'll give it a shot).