What is the motivation behind designing a control stick that does not move?
The stick does not need to move in order for the pilots to sense their inputs!
Humans have very accurate force sensors in their fingers, and no direct position sensors. Without looking, we only have a very vague idea where our hand is due to muscle tone sensing, but we don't need to look at our hand to know exactly how hard it is pushing at an object. For the human, force is the imminent control feedback factor.
The F-16 can do 9G in manoeuvres, and the pilot is inclined backwards with their arm on an armrest. If this arm needs to move a stick it shifts position, and during a manoeuvre can get grabbed by the load factor - precise control is much easier if the hand, wrist and arm can stay in one position.
A pilot who has flown both the F-16 and the centre stick F-18 has this to say about it:

Flying a side-stick control takes a while to get used to, but once you do, it's a joy. The conformai stick's shape feels very natural (it fits in the hand like a melted candy bar), and it allows easy access to nine of the 16 hotAS controls. Two fully adjustable forearm rests on the right cockpit bulkhead stabilize and isolate the pilot's arm and wrist, so when rattling around the cockpit during turbulence or going after the bad guy, the pilot's arm won't accidentally move and initiate unwanted control inputs.

Initially the stick was fixed completely and pilots would get comfortable with the novel concept of no stick movement pretty quickly. But there were two issues:

Cross-coupling. The control input must be separated in two axes, pitch and roll, and this is difficult to separate if feedback is from force only. In order to aid this, a small bit of movement was introduced with a breakout force in the centre: upon release, the stick moves back to neutral and requires a bit of force to be deflected from there. From the article linked above:

However, after initial F-16 flight tests, a ¼ inch of stick movement was incorporated to give a small dead band and a nominal breakout force to give better &quot;feel&quot; of a neutral stick because otherwise it was entirely too sensitive. The control harmony is quite good (the pressures required for pitch and roll mix well), but without the capability to physically position the stick, it's easy to contaminate roll inputs with unwanted pitch inputs, and vice versa


F16s kept getting back with bent sticks. During chasing another plane and pulling, pulling on the stick, they pulled excessively hard while the fly-by-wire was already generating the deflection for maximum load factor. With the stick now moving a little bit, the pilot can feel that the limit is reached when hitting the travel stop at 25 lbs. At this force the acceleration is 9G and pulling harder has no additional effect.