I wonder if competition has some connotations/definitions that make it harder to use in the discussion.
I usually take competition as being something like the OED definition "the striving of two or more for the same object" -- where there are two sides like in football or baseball or tennis. It feels like the DM isn't in competition against the players in that sense, and the players aren't in competition against each other.
Similarly for "Compete - To strive with another, for the attainment of a thing, in doing something." it sounds like the "another" is another team and not a puzzle.
"I competed in puzzle solving" sounds like there are a bunch of different teams doing the same puzzles (like a convention with all kinds of groups going through the same module to see who did it best). It feels odd to me to say "I competed against todays cross-word puzzle".
If it's "struggled to overcome a challenge", does the challenge not have as much connotation of needing a thinking thing on the other side that's trying to beat you?
In a TTRPG, competition can take many, many forms.
* A GM competing with themselves to run a better game than last session.
* A player competing with themselves to thread the skillful and thematic play needle better during this scene than they did in the prior scene.
* A team of players wanting to overcome the obstacle or attrition model for the game's engine + the adventure that they've chosen.
* A pair of players with competing dramatic needs in
this scene playing as thematically hard as they can to see if their dramatic need wins out.
* A GM fairly competing with the players as hard as they can to give them "an honest win" after a conflict.
* A player trying to earn the Teamwork/Sacrifice or MVP or Struggled With Your Belief award at end of session.
* A player trying to harness their "describe to live" better in this session than they did last session (wherein last session, they may have inverted their mental paradigm and started with "how can I marshal as many resources as possible to win this test).
* A player trying to better exemplify a Player's Best Practice in this session (like "don't talk yourself out of fun" or "play your character like a stolen car") better than they did last session...or perhaps exemplify it for the table so others can learn from the Best Practices example (therefore creating a better group of players for this particular game).
* A GM might try to listen better for particular keys from players than they did prior.
* A GM might be putting training wheels on for players new to a system (so they're constantly going over mechanical bits in every decision-point) and they've kind of gotten stuck in a rut of doing this and they need to ease off.
There are dozens of ways that GMs will compete with themselves during a session. Where players will compete with GM content. Where players will compete with themselves (personally). Where players will compete for end of session rewards or will compete when dramatic needs collide in a scene.
Competition is a very very healthy thing between people who respect each other and have empathy and understanding and care for each other. I've competed healthily in physical combat, in dozens of sports, in mental exercises, in gaming, hell...in dusting and yardwork and mundane tasks. Being a better person than you were prior and challenging your friends and loved ones to be better than they were prior isn't a hostile environment. People bring that hostility into it because of their own hubris and deserved lack of credibility and restraint and empathy and understanding when engaging in one crucible of competition or another.