I was gonna agree with you, but then it got confusing when I started overthinking it. The 21st century began in 2001. Year 1 is the first year of ANY century, or millennium, but not a decade?
I think it comes down to how you define decade. If you say "the third decade of the century" (which admittedly nobody does), then 2021 is the first year of that decade. If you say "the Twenties", then 2020 is the first year.
So "they" are not technically wrong, you're just /more/ right.