Most applications don't have to worry about it, but platforms do. The SQL standard requires accurate date representation for years 0000-9999. I was surprised to learn about the Gregorian shift while testing our database system. I also learned about "war time" in WW2 and the "battle of the clocks" in colonial India. A lot of history came out of debugging those tests.
For sure. I was thinking more of "normal" applications, as opposed to things like SQL databases. It looks like the SQL standard explicitly calls for the use of the Gregorian calendar for dates, so I think that answers the previous question of "Who else uses the proleptic Gregorian calendar other than programmers of datetime libraries?"
"normal applications", no. But there are more platforms besides databases, e.g. programming languages. Any platform that works with dates will have to deal with these weird quirks. And application that works with dates probably would not, although there's always leap seconds...