Followed by the well-know theorem:
(A,B) is controllable iff poles of A-BK can be arbitrarily assigned.
(A,B) is stabilizable iff poles of A-BK can be arbitrarily assigned on the LHP
LHP = left-half plane.
My question is we all want the poles of A-BK on the LHP; by doing so the system is stable.
Based on that, it seems redundant for the requirement that (A,B) is controllable, since (A,B) is stabilizable is enough.
I am confused about the hope for controllable of (A,B)
It is obvious that if a system is controllable it is also stabilizable since all eigenvalues can be assigned to LHP. But the stabilizability condition you gave is not accurate since there may exist some eigenvalues that are already in LHP but cannot be arbitrarily assigned. The condition should be that all eigenvalues of $A-BK$ can be assigned to LHP, but not necessarily arbitrarily.