I guess it would be harder to use a single exception callback if you had a lot of similar "try/catch" blocks in different areas of your code, and you'd still probably need to collect some sort of stack trace, but it might work out alright if you go with this article's advice and recommend that people keep exceptions out of their program's normal control flow.
I dunno, maybe it's a dumb idea and I've just been spending too much time with microcontrollers lately. They use interrupt callbacks for all kinds of hardware triggers and software events, and I kind of like how you can compartmentalize error handling into a few predictable event handlers.
That Java stack trace image at the end of the article also brings back some memories...but I'm feeling much better these days.