You are right in the etymology and whatnot of "fan," but that doesn't quite lead to a representative noun. I used the Morgan and Silva Furman University Lexicon to find some words, which I list here in the order they appear in the lexicon, with some added notes:
- admirator studiosissimus: literally "the most eager admirer," I would think this would be better for the phrase "#1 fan" instead of just an ordinary fan
- fautor: a favorer, promoter, partisan, or applauder
The above terms come from the Adumbratio, while the following are from the Silva:
- (fautor) fanaticus: an "enthusiastic one/fan," here we do see the English root return, but as a substantive adjective, or just an adjective
- philathlus: of Greek origin, "one who is fond of games, a competitor," not exactly a fan per se
What's interesting is that admirator does not appear alone, despite meaning "admirer." Perhaps fan is a stronger word, but it depends on what you want to do. Someone who is a fan of art (I'm thinking art as in painting here) is probably not the obnoxious, half-drunken, scream-at-the-top-of-their-lungs sports fan, so maybe this is where such a differentiation would be useful. However, all of the terms from the Adumbratio are not specific to sports, and are therefore the most likely widely applicable terms.