I think the words can be interchangeable and either is often used to mean the same thing in english along with grocery but I suppose it's because there is a Spanish word for market and it is generally a smaller establishment.
Why is "at" the supermarket wrong when "at" is on the dropdown menu for "en"?
At the supermarket = al supermercado
The dropdown menu shows you possible definitions. Think of it as if you were looking up a word in the dictionary. Not all of the possible meanings are the correct meaning for the given context.
It has allowed me tk say grocery store several times now it wants me to say supermarket...
Americans never say "In the supermarket". We'd say "In the store" or "In the grocery store".
…And Americans are obviously the only English speakers. (That is called, sarcasm).