Analog buttons require ADC conversion, which is usually done by successive approximation and takes time.
On most microcontrollers you also only have a single actual ADC and it's multiplexed between the inputs which would further slow it down.
To be clear I have never looked at what's inside a dual shock or looked at the firmware, but my experience with microcontrollers is that digital inputs are orders of magnitude faster than analog ones.
Well, they could have the ADC there and a secondary path that goes around it and the signal is split in two and sent through both, so you get the fast digital imput, but the analogue one is still availiable if required.