My question is, would this have any serious drawbacks, other than loss of anonimity?
The current system of change isn't even very anonymous to begin with. Take a look at any block in blockexplorer, and you'll find that it's terribly easy in most cases to pick out, with reasonable certainty, which output is change and which is a real transfer from A to B. I've boiled it down to a set of three rules that work almost all of the time here:
https://forum.bitcoin.org/index.php?topic=27472.msg346658#msg346658. It's only a matter of time before somebody puts together an alternative block explorer that works around the supposed anonymity of "change" being sent to a new address.