Does an electric stove burner element wear out over time?
They can indeed wear out, though it's much more exciting than what you describe - when the element in my oven (different location, same tech, 40+ years old) blew, it made a lovely arc, and I've heard people describe the same alarming thing happing on the stovetop, sometimes burning a hole in a pot.

I think what you are describing sounds more like the connectors that the elements plug into being dirty, worn, or generally connecting poorly - and once they start doing that, the connector heats up more, and that makes it more prone to corrode more, relaxes the springs, etc. making the problem worse.

There could also be a problem with the "proportional controller" which is what the setting dial adjusts - essentially a timer that sets the element on all the time on high, and turns it on and off when on lower settings, with longer off times and shorter on times at lower settings.