I miss the analog button :(

No, you fucking don't.

Some games turned it on by default, especially on the PS2. Some older games flat out didn't work if you turned it on.

It was for compatibility, kind of like how some Sega genesis games wouldn't work with the 6 button controller until you put it into 3 button mode.

It was basically a compatibility button for older PS1 games that didn't support analog.

What does this button do?

starts the "anal on gamer" functionality

The playstation originally didn't have sticks, see So games released before the sticks weren't programmed with them in mind

Which means, without this toggle, the sticks would be useless for a lot of older games as they don't do anything
But this way, you can map the D-Pad to the left stick

It was for compatibility button for older PS1 games like how some Sega genesis

>that off-white color, almost a beige
peak nostalgia

No, it wasn't compatibility, it's just that not every game is more confortable with the analog stick