You can! Many years ago I took a desktop machine (minus monitor, keyboard, etc as my friends already had spares of these at the destination) on not one but two flights (going there there and the going back back). Surprisingly the people checking in my bag didn't seem to surprised--the only unusual thing was that they asked me to sign a declaration that the airline wouldn't be responsible for any damage. The bag tag went all around the machine (I didn't have the original box so I took a gamble to see if the airline would accept the machine itself) and off it went on the belt at departure. And on arrival it came on the carousel with all the other bags. It worked fine after both trips. All so I could join a gaming session with my friends in another city! Though it should be said I can see why it might be more tricky once you factor in a monitor and other things. And I probably wouldn't do it again now I have a laptop that can run some pretty decent games--but yes good memories... :)
A SFF would do nicely. A NUC even more so, but is not really up to high workloads
Most of these gaming laptops likely don’t; they’re really not intended to be used for heavy gaming while on battery, because the battery can’t support the discharge rate needed by a high end GPU.
Nobody sells laptops with batteries bigger than that! Guess why? Because you're not allowed to take them on planes. Just means gaming laptops have less battery life is all.