I'll let an ex-mechanic answer: me. In that situation the service writer (my go between me and the customer) calls the customer and tells them what is needed. If they don't want to do it, button it up and send them on their way. In your made-up example, there are fixes for a stripped drain plug that don't require dropping the oil pan. If it weren't for Jiffy Lube (who, more than once, stripped it before it got to me), I wouldn't have to know this, but I do. So let's use a real-world example.
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.