Is there a name for that annoyed (yet slightly ashamed for being annoyed) feeling you get when the new and improved way of doing something breaks for some reason and you have to go back to doing it the old way, which worked perfectly well for many years and wasn't really that bad?
Assuming the key features of the feeling are the three given below, you'd be chagrined...
chagrin - a feeling of vexation, marked by disappointment or humiliation
In my experience, when people are described as chagrined it's very often because their bright idea for a new/better way to do something didn't work out, so they've had to admit humiliating defeat. And revert to the old way - bringing even more humiliation, since they were probably complaining at length about the old way before they optimistically tried something new against all advice.
In such a case (and in some others that are different from yours)
it's back to the drawing board.
it's back to square one.
While some may say that these don't name a 'feeling', I do believe these expressions strongly imply the negative feeling you describe. Evidence: "Let's fall back to the original plan" means much the same thing in practical terms, but has no negative valence.