That only works for now because we are so close to 2000. So there is little ambiguity whether "the 2000s" refer to the century or the decade. But in the future, "the 2000s" will refer to the century. Just like the 1900s refer to the 20th century rather than the decade (1900-1909).

Not my problem, that can be fixed by a future generation of pedants. It'll be good enough for 50 years or so.

I would have thought that extends beyond 2009

We don't need to refer to the third millennium as a whole right now, being in the first part of it. And by the time people need to refer to the 2000s, they will no longer have a reason to reference the 2000s.

You can also refer to, in speech, the twenty-hundreds, without ambiguity; I suppose, if you want it more compact in text, you could always write the 20-00s.