Why A320/330 use C* FBW control without speed stability? The A320 and A330 family are FBW aircraft that use C* longitudinal control laws for longitudinal sidestick command tracking. In essence, stick neutral commands 1G flight. When stick is deflected longitudinally, a blend of pitch rate and normal load factor is commanded to provide a consistent G-feel to the pilot; once the stick is released, 1G flight resumes, but the airplane is steady-state at the new pitch angle (or flight path angle to be more exact).

Unlike B777, B787 and A220, however, the A320/330 family do not have artificial speed stability. Once the aircraft is steady-state at the new pitch attitude, and with the autothrottle disengaged, the airspeed will settle to a new trim point. In another word, unlike a conventional aircraft, the A320/330 only has neutral speed stability. 

This is why the envelope protections of the Airbus aircraft are not nice-to-have features, but essential features to ensure the airspeed does not rapidly bleed off to stall or increase to overspeed, all while the stick is neutral. 

My question is, why is a C* control law without speed stability desirable for a transport category aircraft? The A320 does not need to perform precision target tracking as a fighter aircraft would, and it changes the flying characteristics drastically from those of a conventional aircraft.
Many accidents have shown over and over that once the handling becomes muscle memory, the cue that having to pull harder means you are losing speed is easily missed anyway, so it's not as important as it seems.

So the artificial neutral stability instead decouples the controls: stick for flight path angle, thrust levers or A/T for speed. Nice and easy and you don't have to fiddle with the trim button that most pilots end up pressing rather mindlessly.

Large aircraft tend to be flown with auto-thrust all the time anyway, including e.g. a 777 which is stable in pitch and has trim. And when they are, the logic is rather like Airbus—control column for flight path angle, maintain flight path angle when released, just after changing speed you have to fiddle with the trim to get it stable again, which Airbus Normal Law does for you.

Note that aerodynamically all Airbus aircraft are still longitudinally stable, so if the sensors fail and the system degrades to Direct Law, the trim behaves like in any other aircraft.