That's exactly the point. The goal is to make the bitcoins spendable by anybody. This is just a variation of the commonly agreed on hypothetical migration strategy. (Except it would not be voluntary in a sense that bitcoin would remain valuable) It would be the equivalent of throwing bitcoins away in a provable fashion instead of destroying them.
But you can't prove that these are thrown away. You can only prove that they haven't been spent. Thus they simply are non-circulating coins.
[Edit: In re-reading your answer , the "spendable by anybody" is something I glossed over, but after re-reading that I now am not sure I have any clue what you are talking about. I'm guessing that the ten minutes you've put into this idea hasn't been enough time to lead you to the flaws in it that others just assume would have been considered.]
It is possible to verify that some private key received it isn't it? So if this private key is publicized it would be the equivalent of throwing coins away. Anybody who grabs the key and transfers the coins to his wallet first has them after. Of course it could be possible that this way whoever grabs the coins would forgo the possibility of converting his existing coins in that wallet, but I think that depends on how rigorous the implementation is.
Might be similar to the coloured coins proposal.
Wouldn't this coin be dependent in on the bitcoin blockchain, since it's in the business of meddling with bitcoin balances?
Only the conversion part, and if it is really made to be an assimilating currency (lets call it borgcoin) at some point conversion might not function as well any more at the end.
The effect would probably be increasing block intervals due to a rapid drop out of miners and the increasing computational load to determine the legitimacy of the assimilation. I think determining legitimacy is actual a NP-hard problem most likely to be solved with greedy algorithms. But there could be compromises (maybe even nessecary) of lets say only allowing one conversion per wallet. (So there would be no need to determine how many coins were received from publicized private keys)