Because I just reset on my third character (third time on account this season) and even played a match after just in case and nothing has dropped. Its still not selectable in the kiosk even after that and a full game close and relaunch.
If its by character, it should say that. It only says just 3 resets which almost implies they want you to do a reset on each character, hence 3 not being arbitrary but having a purpose.
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Gambit rank is per account, there is no separately tracked Gambit ranks by character.