As it turns out, a mole of carbon doesn't mass exactly 12/16 of a mole of oxygen or 12× a mole of hydrogen. (Carbon's average atomic mass is 12, not 14.) So you'd have to pick an atom.
At present, the definition runs the other direction: a dalton is defined as 1/(Avogadro's number) of a gram. But Avogadro's number is a little difficult to measure directly; what you end up doing is picking an element and measuring its atomic mass. They picked carbon.
It seems to me that the definition you describe would indeed be a reasonable one, since Avogadro's number is known to seven significant figures at this point, which seems comparable to the mass stability of the international prototype kilogram.