I have been helping one kid with a programming class and one thing I've noticed is that they often lack a certain approach to problem solving that I personally developed by playing video games. They also played video games but far fewer and definitely less of the harder problem solving ones. It is entirely possible I liked the games because I already had the talent and the games didn't build the talent, but until we get a definite study on the correlation and causation my bet is going to be on them sharpening the skill.
After a certain point it does begin to sap. This is the nature of all entertainment. A few good books can enlighten a mind but too many books can result in someone who reads but never does anything with the knowledge. Even one doing math for fun can end up spending all their time solving math puzzles that are created just to be math puzzles and which don't tie into any greater discoveries.
I think this is fine. First, the system is far too hard to control so I don't see any way to improve it that wouldn't wreck the system. Maybe minor suggestions to diversify entertainment a bit and to be sure to share your experiences with a like minded community, but these are small tweaks and not massive overhauls.
I also don't think we can predict when something will produce the best results. Even if 9 of every 10 kids who get heavily drawn into math problem solving end up just wasting time solving math problems that exist just for the sake of math problems, that last 1 might get into applying the problem solving to novel fields of math that expand the total knowledge of humanity. If one is too heavy handed in optimizing that they'll risk destroying the ability for that 1 who goes further. I doubt we can be sure that trade off is worth while.