Certainly before Obamacare, private insurance companies spent about 30% on overhead (enrolling & dropping people, advertising, dealing with referrals, arguing with hospitals and doctors, auditing people, hiring lawyers to get someone else's insurance to pay for treatment). That's a lot of money and it figures into "healthcare costs" in the US.
We spend so much more than any other country on healthcare, so much more.
You are absolutely right that "the current system needs to collapse under its own weight before a new approach even stands a chance". I believe it will take people dying publicly to make the US consider a change. First we will be bamboozled by the "freedom" and "choice" we'll get with the new plan, foremost the choice not to have coverage or care at all :)
Note that I'm not saying that we can't eventually devise a market-like system for allocating medical resources. I do wonder if we have the technology today for building such a system though. After all, we'd ideally have very good information very quickly so that we can make decisions about where to spend. I'm just not sure we have anything close to such a thing yet, and perhaps we should stop pretending that we've got one.
Casinos use chips and vouchers to distract gamblers from actually grasping just how much money they're playing with. Health insurance sure feels like the same thing, but it's even worse in that when you're in need, you're not always in a state of mind to make rational choices nor are those choices even adequately defined prior to receiving a bill. Why is this acceptable? Since when are contracts entered into under duress admissible? I suspect it's probably a massive success on the part of lobbyists.
I don't know the solution, but I can confidently say it's probably not reached by tackling the insurance side of healthcare. Maybe the current system needs to collapse under its own weight before a new approach even stands a chance? Unfortunately, it's probably more likely it'd just receive a government bailout at taxpayers' expense so it could continue the status quo.