[Hockey Stick] No
[Essential] After quite a lot of thought, No. It's quite important, but not as essential as, say, the ball in football.
[uniquely used] I'm not 100% sure what you mean, but the AOTC is not used in any non-team variant of the sport(s).
[rethinking] No. The circumstances under which two hands are used are a giveaway.
And now you're all just toying with me, I think.