Ignoring the safety of the people doing the fixing for a moment, how do we know that these potholes have been fixed in a way that'll last a long time and be safe? If I want to know the method and materials used, where do I go to find that out?
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).
Many local governments grab any able bodied people to shovel semi-hot asphalt into the holes and let the traffic pack it in. This is not the recommended procedure by any standard, but people have gotten used to the incompetence and actually believe that this is proper, just because someone official is doing it.
It doesn't take rocket science to shovel in a bunch of cold-patch and sand and tamp it down.
And that is what they do, unless they are completely repaving the road.