Some features aren't available on some device running One UI Core. If I remember it right, the A10s is running on it, that might be a reason why you can't find it even in the Galaxy Store.
Some OneUI Core devices actually have Game Launcher pre-installed. I saw the Game Launcher on the A12 despite running OneUI Core.
I believe it's hardware-dependent. If the processor is "gaming-capable", it might have a Game Launcher.
If you install games from the Galaxy Store, they may run smoother than ones released in Google Play Store because they are optimized specially for Galaxy devices
