This whole idea about anxiety is overblown. It always get reducted to how good you are. It maybe counts as something like 5%. A great player who is shy and has anxiety problems is going to always be better than an average player who doesn't. It's just how it is. Anxiety manifests and is a big weakness when you are facing things you are not comfortable with. But when you are good at your job, its effects are minimal.
Some of the best in their fields were shy people and could pass off as people dealing with some forms of anxiety. It means nothing when you're good. By the same logic, many worldwide superstars confessed to feeling nervous before concerts. What matters most is to perfect yourself, and evolve, and not let the emotions get in the way of performing at the level you prepare at. The weaker who can't do that don't fail because of anxiety. There's always bigger flaws that get in the way than only this.
This is true, sometimes the effect of 'mentality' is very overplayed.
Sometimes, rarely, it can stop a player from performing if it's rising to the level of a serious medical problem, but in these cases, there is usually something you can do about it.
In general, a player's quality will usually shine through regardless, their level might reduce, but you will still be able to see they are a good and talented player.
Its mostly down to ability.