A surprising number of words arise from a process called “misdivision”. In this case, “-gate” was coopted as a generic suffix for scandals, in the same way that “-burger” is a generic suffix for certain types of hot sandwich, even though the name comes from Hamburg+er, not ham+burger.

Off the top of my head: “a napkin” used to be “an apkin”, “a lute” used to be “al ’ud” (“the lute”, in Arabic); “an orange” used to be “a norange”, and moreover, the colour is named after the fruit—we used to call the colour “yellowred” (geoluread, in Old English).

Language evolves in messy and illogical ways.