Ellsworth A320 full back stick
Which would mean if you pulled the stick fully back, the aircraft will NOT give you Vx. Is that Correct?
Correct. The result is to give you a coeficient of lift very close to the maximum one (top of the curve on the CL/AOA curve).
Then howcome in GPWS Pull-up warning (in which you want Vx, not Vy), the procedure is TOGA, stick fully back and let the speed drop all the way to Valpha max? Isn't that giving you less-than-maximum climb angle?
During such a situation your goal is to obtain maximum aerodynamic lift. What you do is changing your kinetic to potential energy up to the "alpha" max. By then you should be free from troubles coming from EGPWS terrain closure.
All the best