I firmly believe we'd see a lot of problems. Pick nearly any datetime library and see how easy it would be to calculate things like durations between two arbitrary timestamps. Even with those that do handle daylight saving or time zones properly, many applications that do use these libraries don't use these features. Or how often projects update their timezone databases. Handled in an incremental way would be a world of pain, precisely because of our reliance on modern electronics and software, or more precisely, because we rely on time so heavily to coordinate events, and rely on syncing these operations with human activities. I think I may have just literally shuddered.

By the way, what you describe is how things worked prior to standardization of time which was driven in part by the expansion of the railroad. There's a brief history on Wikipedia:

https://en.wikipedia.org/wiki/Time_zone#History