The current system was built for agriculture - "quarters" matching periods of different agricultural activities. It just does not make any sense in post-agricultural societies. This was clearly recognised at the beginning of the XX century, then abandoned as anglo-european elites slowly retreated into cultural traditionalism (as a defensive reaction against the rise of socialism).

Saying that it doesn't matter because computers do it for us is just hiding dirt under the carpet. I agree that timezone and second adjustments are more problematic, but having a more rational overall approach would also help there: you would have a "hack zone" where to dump all the madness (the special 1-day/2-days period), making it easier to deal with any adjustment. Currently, you are inserting most changes on 1 January and / or 31 December, but then also hacking February for no particular reason, all the while not being able to use the week as a meaningful unit.

Throughout history, we experienced periods when people would not shy away from bold cultural and rational changes in order to improve life for future generations. Holding the (adjusted) Gregorian calendar as an ultimate taboo is just sad.