the idea that the year 2000 is the start of the 21st century is hurting my head.
That may be because it is not. The first century was years 1 to 100. The second was 101 to 200. The 21st is therefore 2001 to 2100.
What you’re probably referring to is the “cultural century” which was considered to have started when the lead digit changed from 1 to 2. The same thing happened quite recently when some people argued 2020 was the start of a new decade (again, it wasn’t)