Yes it does because the mechanic is required to call you and say, "Upon further examination we also need to do ______ with will cost $XYZ. Do you want to proceed with this?".
If you say no you can pick up your car and go somewhere else. It's not that the mechanic is taking on the risk of an oil change turning into a full engine rebuild. They're simply required to announce what they're going to bill you for before they start work.
That way if the mechanic say, "It'll be $500 for me to change your cabin air filter." you can find a new mechanic.
But, that idea ignores a fundamental problem with expert-based markets; it's impossible to me, as a non-consumer, to make a rational decision about which mechanic to use because to me, the mechanic is a black box. I don't have any way to immediately fact-check what they're telling me, and in the worst case, if a mechanic does something wrong, I may not find out about it until it's far to late to hold the do-er accountable. Thus, I can change mechanics based on the limited information I have available (price signals, perceived reputation) but, A) this distorts the market to optimize for low prices and perceived reputations instead of efficient prices for quality work, and B) I don't ultimately have any way of knowing whether I'm making the right decision when participating in this market. At least, not in the time frame where that knowledge is useful.
The same, holds true with 'free' health markets; I'm not a doctor so I can't accurately tell if my doctor is lying to me, which should cause similar distortions in an ostensibly free healthcare market. Additionally, as with an auto mechanic, if I need something done -now-, my demand for that service becomes completely inelastic, which means prices for emergency services/procedures should become obscenely high.
They already are today, and people are -already- getting forced into bankruptcy by emergency medical expenses, so I can't speak to whether or how a market sans insurance might change that phenomenon, but it seems to me that the goal ought to be to eliminate it entirely, and that in turn a 'free' healthcare market cannot be the end goal.
That said, I don't think I know enough about the subject to pick out a particular optimal solution to this problem. I like the -idea- of single payer or other universal healthcare solutions, but I haven't studied any of them in enough depth to analyse them critically; dealing with healthcare in the UK was a joy while I lived there, so I wouldn't mind having that experience here in the US, but that in turn requires a lot of government funding (among other things), and we can already see that that's unsustainable in the short term by what's happening to the ACA.
Customer comes in for an oil change. Change the oil, go to start the car, it won't start. A bit of diagnosis later, and the ignition control module has coincidentally decided to die. It's an old GM, so thankfully it's only $35, but still: the car was running when the customer dropped it off. I hate those phone calls. Customer said, "okay, do it". Had they not, we probably would have just dropped a module in and eaten the cost. I mean, what are you going to say? "Umm, yeah, it's just coincidence that you need to spend $35 more before you can drive it home"?
I also tell customers ahead of time the possible outcomes of the work. Honda comes in needing a top-end job, but it has over 100K miles on it. I tell the service writer that we can do the job, but it stands a chance of blowing the rings. (New valves with better sealing on the top end, now the piston rings are the weak link.) Customer is informed, I do the job. Two weeks later, the Honda pulls up blowing smoke. Told you so. I forget the outcome, but I don't recall pulling the engine on that one.
But they don't always say, "yes". Customer comes in looking for an estimate on a brake job. You know that grinding you hear when your pads are shot to hell? Imagine what happens when you let that go for, say, a couple of months. Imagine no more, I'll tell you what happens: it'll grind the rotors down to the cooling vents such that what's left of the brake pad backing plate no longer has a solid surface upon which to grind. Of course wheel bearings are shot, calipers need replaced. It was a long time ago, but many hundreds of dollars were estimated. Of course this is on a piece of shit that isn't worth fixing for that much. The vehicle is not safe to drive by any means. Customer doesn't want to do the work (duh). What did I do? Put the wheels back on and let them drive off. We're mechanics, not the state police, but trust that there was discussion of what "the right thing to do" was.
Your surgeon is in the middle of heart surgery. You wanted an estimate on a (cardio) valve job. Surgeon gets in there, your arteries are all clogged up. Wake the patient and ask for approval on the extra expense? Button 'em back up and negotiate the cost later?
So the car analogy doesn't work for all cases, or very well at for human beings.
Many medical procedures and therapies have very predictable costs and outcomes. And have checkpoints to re-evaluate whether to proceed further.
And accountants and could probably also price in the cost of the occasional problematic case and include a buffer in the standard price to account for it.
There are plenty of straight forward ways to provide prices for the vast majority of medical procedures, if the proper incentives were in place.
Does that include a situation where say, when emptying out the oil, the nut over the drain breaks off which necessitates changing the oil sump, except they then discover the engine mounts are rusted and now the engine needs to be lifted and the mounts replaced? (I'm not a car guy, but I hope this demonstrates the thought even if inaccurate)