Look at it via the basic equations for flight performance for unaccelerated flight. T= thrust, D = drag, L = lift, W = weight.
T = D + W * sin(flight_path_angle)
L = W * cos(flight_path_angle)
Thus
(T - D)/L = tan(flight_path_angle)
The maximum flight_path_angle is at maximum (T - D)/L, i.e. at maximum excess thrust. For T = 0, the maximum flight_path_angle is best glide angle and reduces to maximum -D/L, which is equivalent to maximum L/D.
Multiply through by airspeed v to make the right hand side vertical velocity and get:
(T - D)*v/L = rate_of_climb
Thus best rate of climb occurs at maximum excess power and for the T = 0 case, best rate of climb is minimum rate of descent and reduces to maximum -D*v/L, which is equivalent minimum power required. This is unrelated to stall speed, though it's possible in some cases that it might occur very close to stall.
Maximum lift as an absolute quantity for unaccelerated (1G) flight will indeed occur at minimum flight path angle, i.e. level flight if you have the power to stay level, but it's not a very useful concept. Maximum lift coefficient does indeed occur at stall, but by the time you've reduced the speed to that point, it's not the same as maximum lift.