No, a stick is pivoted at the floor - correct? And the hand grips the stick in the palm with your fingers wrapped around - right? This has a totally different movement and total throw, compared to a joystick type controller. The total throw is what... 3 inches max forward/back for the joystick. Very different and uses a completely different set of muscles.
I guess there are just folks that refuse to see to control for what it is - there is also a wrist rest - the arm doesn't move, only the wrist and fingers.
Not all sticks are pivoted at the floor. I've flown several (most) that had the pitch pivot at the floor, but the roll pivot higher on the stick. Also, floor height may be significantly different among types.
And, NO, I do not grip a stick in the palm! Depending on the airplane, different parts of the FINGERS are used to grip the stick (and occasionally braced with the edge of the palm), but SELDOM with the fingers wrapped around!
Regardless of the total throw, the grip on either a conventional center stick or a sidestick is VERY dependent on the individual installation. You apparently have little or no experience flying real airplanes with stick controls...
While there may be a wrist rest in SOME airplanes, and SOME sidesticks use no arm movement, that is definitely NOT a universal truth! A brace for the forearm or elbow may well replace a wrist rest. I doubt you could fly a Cobra without arm movement, though it may be possible in an F-16 or A3xx (I've flown neither of the last 2).