The honest answer is that Rock Paper Scissors is a game first. The sport argument only begins once you add recurring tournaments, referees, clear rules, and players who practice enough to improve. Some communities have built exactly that kind of structure around RPS, which is why the question refuses to go away.
The case for calling it a sport is straightforward. Competitive players study tendencies, develop counters, and train themselves to avoid obvious patterns. Refereed matches care about timing, illegal throws, and repeatable rules. Practice changes results, and once practice changes results, people stop treating the game as pure luck.
The case against the label is also straightforward. The game is fast, visually simple, and not physically demanding in the way most people expect from sport. That makes the word feel inflated to some audiences, even when the competition itself is real.
So the better answer may be that RPS sits in the same family as other organized mind-and-pattern games. If you mean "sport" as something that requires structured competition, officiating, preparation, and a measurable edge from practice, the argument is plausible. If you mean a physically demanding athletic contest, the label is harder to defend. The interesting part is not the vocabulary. The interesting part is that the game becomes much more serious once people start training for it.