Is there a good test for whether a flat-top stove is working correctly?

This is, indeed, how electric stoves (flat top, coil top, or otherwise) work. It's also how most microwaves work, incidentally. The heating elements have two states: on or off. If you set a burner to an intermediate setting, the way it achieves that intermediate temperature is by turning on for a particular amount of time, then turning off for a particular amount of time.

Note that the burner doesn't actually cool down all that quickly just because it momentarily doesn't appear red. This is why it's so easy to burn things on an electric cooktop if you're not used to it: burners stay hot enough to cook food for quite a long time after being turned off. If you're done cooking something, you need to physically move it off the stove, or at least off of any burner that has been turned on in the last, oh, half an hour, otherwise you will continue cooking it, thus potentially burning it.

To test whether a burner is working, turn it to the highest heat and check that it stays on more or less constantly. Note that depending on your stove, you might have to put a pot on the burner for it to turn on at all. And if your stove comes equipped with a "boil-over" sensor, all bets are off: the burner may not stay on constantly no matter what you do. In any case, the failure state of most stoves is "stops working at all", not some sort of "works when it feels like it", so if you can get your stove to produce heat, chances are it's working as correctly as it's capable of.

A simple and straightforward test would be to put a measured amount of water (e.g.: 4 cups) in a pot and place it on a burner turned to the highest setting. While being careful not to watch, time how long it takes to reach a full boil.

Repeat the process on the other burners, making sure that you use the same amount of water, in the same pan, and starting at the same initial temperature each time. If the burners are all the same size, it should take approximately the same length of time for each to reach a boil.

Of course, if the burners are of differing sizes there'll be a variance, with larger burners working faster. Otherwise, the times should be consistent. If there's a marked difference however, you've pretty much proven the burner(s) aren't working properly.

Note that if instead of conventional resistance heating elements, you have an induction range, there could be other issues at play. Make sure your pans are compatible with induction elements. Otherwise, the control system could be detecting a problem and causing a shutdown.

As others have said, it sounds like your stove top is behaving normally.

I am guessing that you have very thin aluminum pans. This allows the pan to reach the full heat temperature during the stove top's heating cycle and this will burn your food. Am I correct about that?

If so, I recommend you buy some sturdier cooking ware. You don't have to buy very expensive stuff, just some stuff that has more bulk to it on the bottom so the heat doesn't transfer instantly through it. You want your cookware to store some of the heat and release it evenly to your food. If you have the money, a decent starter set is that pioneer woman $119 starter set that includes some non-stick aluminum skillet, a cast iron skillet, small stock pot and sauce pans.

If you can only afford one pan at a time, get a skillet like calephon or similar make of pan. This will help your cooking out tremendously.