I've done it both ways (carry-on vs. checked) and have had no issues. Either way, I like to remove the head. My old head was a "regular" 3-way head with three longish knobs; I'd unscrew them as well from the rest of the head. Saved space that way. My ballhead, though, has nothing that can be disassembled and sometimes I wonder what it looks like on the x-ray! Probably like a little grenade or something. No issues though, knock on wood.

And, yeah, I go with the big & heavy tripods. The tiny lightweight models aren't too stable with a heavy DSLR with vertical grip and long zoom even when used in perfect conditions (like indoors); I can't imagine how unstable they must be outside in high winds especially if I need to extend it all the way up. No thanks.