Look some info on BCD or EBCDIC.

Oh boy you heavily underestimate the amount and level of bad decision in legacy protokoll. Read up in the toppic. the Date was for a loong time stored as 6 decimal numbers.

With an 8-bit signed integer counting from 1900...

Some of the computers in question predate standardizing on 8 bits to the byte. You've got a whole post here of bad assumptions about how things worked.

You're thinking of the problem with modern solutions in mind. Y2K originates from punch cards where everything was stored in characters. To save space only the last 2 digits of the year because back then you didn't need to store the 19 of year 19xx. The technique of storing data stayed the same for a long time despite technology advancing beyond punch cards. The assumption that it's always 19xx caused the Y2K bug because once it overflows to 00 the system doesn't know if it's 1900 or 2000.

a whole lot of programmers would have had to go out of their way to be really, really bad at their jobs.

You don't spend much time around them, do you?

You do realize that "counting from 1900" meant storing only the last two digits and just hardcoding the programs to print"19" in front of it in those days? At best, an overflow would lead to 19100, 1910 or 1900, depending on the print routines.

And then there is PIC 99 in Cobol. In modern languages, it makes no sense, but there is still a lot of really old code around and not everything is twos complement, especially if you do not need the efficiency in memory and calculations.

The issue wasn't using the dates. The issue was the computer believing it was now on those dates.

I'm going to assume you aren't old enough to remember, but the "only two digits to represent the year" issue predates computers. Lots of paper forms just gave two digits. And a lot of early computer work was just digitising paper forms.