When I read the "autopilots can break" rationalization, I know for a fact that the reason they are so often inop or impaired is because they are (or can be) difficult and expensive to maintain and owners simply won't spend the money to find the right shop to fix them properly. Pilots in the part of the country know that Autopilots Central can repair and maintain almost any GA box, so the first question asked about balky AP's is "have you talked to Ferguson about it?"
Being able to properly set up the GPS for coupled AP approaches is one of the most important functions for single-pilot IFR, and I can count on one or two fingers the number of times that pilots have been able to do so (the first time) during FR's.
In short, I've been watching this mess from all of the seats (left, right and instructor panel) for more years than I care to think about and am convinced that we aren't there yet in terms of providing the training necessary to solve the problem.
Ours is busted because even in tip-top shape, it's a POS. Doesn't even have a heading bug. I've spent a bit of time figuring out how it behaves and about all it's good for is asking for a vector and a climb and setting it to hold something like a 10 degree wide general heading and get your bearings if you were seriously impaired. For that role, I'd use it in a horrible pinch. Frankly, trimmed... The airplane does as well or better on its own, left to its own devices.
We've spent a little money on it to get it to that point, but even if it were flawless as it came from the factory, it'd still only be good enough to do that. I let it attempt to intercept a Localizer once, and repeated that just to prove it really was as awful as it was the first time and decided it would do an excellent job of killing everyone on board.
I've also played with it in cruise. It'll keep the airplane upright. Barely. Long enough to use it when reaching for something in the back. It can lower workload so you can maybe prep for an approach but I won't use it down low in the weeds.
I think we could take it to a place like you recommend and it'd go from "dangerous" to "flies like a drunken sailor".
I can't imagine anyone ever used a Cessna 200 to really fly hard IMC for hours on end. They're just not good enough at doing what they were intended to do. None that I've ever seen operable, anyway. Cessna 300 is better with a heading bug, at least if there's vacuum and no DG problems, it'll hold a heading.
Your comments about folks not knowing how to use their tools when they have a good one, is VERY well received here. I can't imagine not knowing how to use the tools at hand. That drives me batty. If ours were useful, I'd be using it, but I'd do a lot of practice with it both operable and in-op.
As far as the long trips go... One has to know their limitations. Multiple hours in IMC hand-flying is fatiguing and knowing when to land and stop is as important as knowing what the airplane can do. None of my trips are "must get there" trips for that very reason. I'd have to feel like today was a tip-top day to file and fly a multi-hour IMC flight with no breaks anywhere along the way in my bird.
My airplane is capable of IFR. Sometimes the pilot isn't. The pilot is a go/no-go item too, just like the airplane. If the pilot ain't up to the task TODAY, relying on an AP to fix that, is just asking for trouble. It frees up brain cells to get the pilot back up to "if everything is going right we will survive this" level, and paints a trap for when it craps out and the pilot isn't ready or able to be the pilot.