It can look dumb, but I always had this question as a kid, what physical principles would prevent this?
If you're openminded enough to listen to those who disagree with the standard model,
take an elastic band and turn one end. Instead of the band turning, you'll have a twist in your band
and it takes time to unravel the twist if you let go on the other end.
That's what will happen to the stick and this travels at lightspeed,
because this is how light works. Light works like 'the stick' in your example.
And if you try turning it faster the 'elastic band'/stick/'atom on the other end' starts breaking.
If you need FTL communication, then use gravity..somehow.