It's possible that an incorrectly fixed pothole would a) degrade far more quickly and b) do so in a manner that expands the pothole more than just leaving it alone would have.
That said, addressing this and other concerns isn't rocket science. The city could easily publish a detailed guide to filling potholes (what materials are ok, what materials are not, what process to use, etc).
It's much easier to just allow citizens to call a number and report a pothole. Seattle does that and potholes get filled fairly quickly, they even have a phone app that allows you to report all kinds of problems.
Here's Seattle's pothole map showing where and when they have been fixed.
We must have different definitions of quickly! Or perhaps different standards for “pothole”.
Council crews everywhere could realistically have best-practice blogs detailing all sorts of stuff to encourage residents to aim for a certain standard (verge upkeep, etc) or achieve other targets (water usage and the like).