It wouldn't be atomic, and so would break transaction semantics.

If you committed a row update but didn't update the index, a subsequent query using the not yet updated index would not find the updated row correctly.

It would also only work for certain types of indexes, you couldn't do it for uniqueness constraint for example.

I do agree that in theory you could have some extension to the index declaration that covers all that, but my worry there would be that it would be non obvious and a foot gun. Doing it the way described above makes that break in semantics clear.