Many mathematicians do in fact teach the rules of the game in numerous introductory texts. However, you don't expect to have to explain the rules every time you play the game with people who you've established know the game. Any session would take ages if so, and in many cases the game only become more fun the more fluent the players are.

I'm not fully convinced the article makes the claim that jargon, per se, is what needs to change nor that the use of jargon causes gatekeeping. I read more about being about the inherent challenges of presenting more complicated ideas, with or without jargon and the pursuit of better methods, which themselves might actually depend on more jargon in some cases (to abstract away and offload the cognitive costs of constantly spelling out definitions). Giving a good name to something is often a really powerful way to lower the cognitive costs of arguments employing the names concept. Theoretics in large part is the hunt for good names for things and the relationships between them.

You'd be hard pressed to find a single human endeavor that does not employ jargon in some fashion. Half the point of my example was to show that you cannot escape jargon and "gatekeeping" even in something as silly and fun as a card game.