First up, MCAS is not a stick pusher, the stick pusher is a separate system that literally shoves the stick forward just prior to a stall.
As for why have a stick pusher at all, first you need to look at the stalling characteristics of traditional straight wing aircraft. What happens in a docile C172 as it stalls is that you first get some buffet as turbulent air off the wings interacts with the ailerons and elevator and just generally vibrates the airframe. Next, as the stall is entered fully, the point on the wing the lift force acts through, the centre of pressure, moves aft. This causes the aircraft to pitch nose down which pretty much breaks the stall unless the pilot holds the yoke back. So you have the stall warning, the buffet, to indicate the aircraft is about to stall, and then you get the stall identification, the dropping of the nose, as the stall occurs. This is very nice behaviour that requires no special mechanical features.
As aircraft became bigger with artificial feel in the controls and swept wings, the traditional stall characteristics were found to be either missing completely, not sufficiently pronounced, or even reversed. In larger aircraft, including straight wing turbo-props, there is not enough natural buffet felt through the airframe or the controls to give sufficient warning of an impending stall. The stick shaker is designed to mimic that buffet, it is a stall warning system. Jets with swept wings were also found to either not have the nose down pitch associated with the stall itself or they would do worse things such as pitch up and deepen the stall. Aircraft like this that showed adverse stalling characteristics were fitted with a stick pusher which does two things; it mimics the nose down stall identification of a traditional straight wing aircraft, and it forces a recovery prior to entering the stall. The pusher is a stall identification system.
So you see the shaker and pusher serve different purposes. Not all aircraft with a shaker have a pusher. The smaller Dash 8 turboprops only have a shaker, however the larger 300 and 400 series Dash 8s have a shaker and a pusher. The B747 had such good handling characteristics that it did not have a stick pusher except for some certified in the UK that had a pusher fitted because the UK test pilots were unhappy with its stalling and thought it needed one. Airbus FBW aircraft don’t have a shaker or a pusher. In normal law the flight computers literally won’t let it stall while in the degraded laws the stall warning system is an aural alarm “STALL STALL!” and there is no stall identification as such, just the warning.
The MCAS is different. I’m not an expert on it at all, but I understand the new LEAP engines fitted to the B737 Max had to be moved further forward and higher on the pylon compared to older engines due to the short undercarriage so that there was enough ground clearance. This new engine position produced some odd aerodynamic effects at high angle of attack that resulted in the nose pitching up. The aircraft could not be certified like that and the fix was to fit a system that trimmed the nose forwards at high AoA to counter the nose up pitching moment of the engines. It’s not a replacement for the pusher.
The Airbus A320 NEO, which also has CFM LEAP engines as an option, already had good ground clearance so they didn’t have the issues trying to fit the large engines on to the existing airframe.
