Doesn't this eliminate some of the anonymity of the system by advertising who the receiver of a transaction is?
No.
Input: 5 BTC to 1someaddress
Output: 4.99 BTC to 1someaddress
Net fee: 0.01 BTC to miner
Maybe:
Original transaction:
Input: 4.99 BTC to 1someaddress from Alice
Output: 4.99 BTC to 1someaddress
Appended transaction
Input: 4.99 BTC to 1someaddress from Alice
appended Input: 0.01 BTC from someone who really wants to get these 4.99 BTC to 1someaddress, presumably the address owner
total new Input: 5.00 BTC to 1someaddress
Output: 4.99 BTC to 1someaddress
Net fee: 0.01 BTC to miner
Only the person who controls 1someaddress can create such a transaction, so there's no reason for him to add an outside input - unless it's a regular transaction going to someone else (in which case the same privacy methods apply).