OG Dualshock 2 never had drift due to internal hardware/software that negated worn down potentiometers.
Dualshock 3 got rid of this expensive solution and we had drift controllers ever since.
that's not the kind of drift that is the issue, also you are wrong as a side note... the PS2 did nothing special to mitigate drift.
as to why you are wrong:
all the PS2, or more precisely PS2 games, do is reset the stick center after some or each load screen.
all that does is readjust the deadzone, which was HUGE btw. and wouldn't be acceptable for our modern standards, and that's all it did...
that does only help with the drift caused by a worn out spring, and it only does so in a very crude and haphazard way
the kind of drift that this "stops" is caused by worn out springs that don't center the stick correctly. that kind of drift is not fully fixable in any way unless you replace your springs.
the kind of drift that is the real issue is worn out conductive material that fails to detect inputs or, even worse, splinters off and gives constant unwanted inputs.
and no PS2, PS1, og Xbox or whatever controller is immune to that. only Hall effect sensor sticks are, like the one in the Dreamcast controller.
no controller is save from worn out springs tho.