Maybe if the miner had no way to climb out he'd stay in the hole and finish the job?
Nope. The problem isn't that the miner is *choosing* to climb out of the hole (they climb out as freely as they climb back in, that is, never), it's that channeling under himself causes the miner to fall, a falling dwarf will attempt a wall-grab, a successful wall-grab results in a climbing dwarf, and a climbing dwarf will not path to jobs (they seem to path to a "safe" location, possibly a meeting area or their previous-idle location).
You can work around this issue by using up/down staircases instead of vertical shafts in most cases. Water will flow freely down a staircase, and refuse, minecarts, and even prisoners will fall down as well (prisoners have a fair chance of grabbing the walls, which is true in the case of a single-tile shaft too). You do need one tile of "free fall", so channel out the top of the stairway after it has been dug.
If you absolutely want a vertical shaft, you can dig channels in a 2x1 pattern, alternating which tile on each level. This will prevent the miner from falling, thus forcing him to complete the job just like in the old days. Then a simple cave-in to remove the ramps left will create a 2x1 vertical shaft straight down.
In this instance, it was a garbage chute, so stairs aren't an option.
Looking again, I think what may have made the difference later was that I had to build a wall around it in cavern 1. Once I was past that, the dorfs didn't (couldn't?) climb out anymore.
The incinerator is now operational.
Thanks for the analysis of what goes on, that makes perfect sense.