I'm not sure I entirely agree: curiosity and discovery can be very enjoyable; which means it is more about openness -- personally I over analyze when I am contracted and anxious, which figures.
The question is still there: how do you maintain openness in a high-stake situation? When you play an instrument in front of a large audience, without enough practice it's hard to be open because all your attention will be directed to get things correct. With enough practice, your muscle memory will handle the basics, upon which openness/enjoyment can be built.
I think a person who overthinks does it regardless of the stakes. Focus is very closely related to not getting interrupted by your own thoughts. I agree with you that you need to have the skill to be able to perform well, but I'm suggesting that even in a low-stake one, one who can't focus won't focus, be it overthinking or distractions.
I learned this from this brain exercise game, Peak. I started leveling up like crazy when I stopped thinking and speaking to myself and let my brain do what it does best, which learning.