There absolutely is a zeroth second! It is the 60th second of the prior minute. This is actually the exact reason why midnight is 12 AM not 12 PM, and noon 12 PM not 12 AM.
If there weren't a zeroth second, minute, hour, day, month, year, or century when was 12:00 AM of Jan 1, 2000?
This is why a zero _offset_ is the initial _value_ for many implementations.
... And I (a non-native English speaker living in the US) always have trouble remembering what 12 AM and 12 PM means, even after all these years I have to look it up every time, and sometimes it drives me nuts.
(For a moment I hoped your explanation would at least serve as a heuristics... but alas, it does not make sense to me. Well, at least at the moment I remember the notation).
Regardless of it, "12:00 AM of Jan 1, 2000" is not a particular second. It's a point in time, it has no duration.
It does not belong to any particular calendar hour or day or year, it's a _boundary_ between two.
Of course it belongs to XX century since XXI century starts at 12:00 AM of Jan 1, 2001.
The first second of Jan 1 lasts from 12:00::00 AM to 12:00:01 AM.
First minute of that day lasts from 12:00 AM to 12:01 AM.
Etc.
If there weren't a zeroth second, minute, hour, day, month, year, or century when was 12:00 AM of Jan 1, 2000?
This is why a zero _offset_ is the initial _value_ for many implementations.