In fairness, you'll have the same problem with solvent cements - they are designed to be a solvent on the native material, not the dried glue itself. If you want to get a firm bond from a broken joint you really need to scrape clean the original joint down to plastic if you can. When I'm making molds the air bleed lines are typically .035 Evergreen rod, lightly glued to the surface of the master. They frequently break off when the mold it first removed from the master and have to be re-applied to make another mold. SOP is to clean both surfaces as simply doing more glue on the original joint doesn't work very well at all.
My first experience with the 'miracle' of ACC was back in the 1970's when it first came out. I had a snowmobile, and the glass headlight bulb actually broke out of the brass socket, yet was still working just dangling on the filament wires. I got my 'model train' new ACC out and tried to glue the bulb back in the socket. It actually worked. And it never came apart over several more years, machine was sold with that light bulb still glued together. Amazing stuff.