Yes, the fact there aren't tournaments with cash prizes for the winners does go a long way to showing they aren't competitive games.
Huh? Do your play bridge or 500? The partners aren't in competition - they are cooperating in the bidding, and in 500 are cooperating in the play.
The fact that they're cooperating doesn't change the fact that questions of player skill, and how to handle it in a social context, come up. And it's not as simple as just allowing the more timid player some spotlight time. I've played with partners who
want me to steer the bidding for our partnership, because they (i) want the pleasure of winning, and (ii) want to learn how to play the game, and therefore are happy to have a model.
In the RPG context, the analogue is that of letting the skilled wargamer make suggestions about spell load-out; or about combat tactics. That happens from time-to-time in RPGing, and has nothing to do with whether or not the game is competitive.
However, as DM, you should take some effort to shine the spotlight on the less skilled players. Or accept that you're actually playing a competitive game.
Why is it up to the GM and not up to the other players? Or up to the less skilled player to make some sort of move?
The GM has a a certain role in the way the game unfolds, but isn't a chaperone. The GM can frame situations that speak to the concerns of the less-skilled player, but if that player doesn't follow up (because s/he doesn't know how to, or is to scared to) then the focus is fairly quickly going to swing back onto a player who is ready to engage the game. And that has nothing to do with competition.
(Of course you can avoid the above dynamic by avoiding player impact on the shape and direction of the game, and just having the GM tell the players stories about their PCs. But personally I don't like that style of RPGing.)