I can see why moving fiat P2P is hard, but why impossible?

It seems you could at least have meeting-less localbitcoins.com as long as

• you can set up a fiat money transfer with unique ID that states where the money is to go
• you can publicly prove you initiated the ID'd transfer (fiat money on its way, shown on bank website, etc.)
• failure to initiate the transfer within a certain number of minutes/hours would also be public

If Alice wishes to purchase BTC from Bob, the process for a trustless, meeting-less trade would be:

1) Alice draws up a money transfer ID at bank, etc., and informs Bob of the details but does not actually send it yet
2) Bob puts the BTC into escrow with the P2P software, which uses an oracle to monitor the bank website for posting/cancellation of the transaction
3) If the transaction is posted (because Alice actually initiated the transfer), the oracle releases the BTC from escrow to Alice
4) If the transaction ID times out, the oracle releases the BTC back to Bob

Alice can't cheat, Bob can't cheat, Alice can't lose, Bob can't lose. The coins only move if the fiat moves. Note how the bank is unwittingly the trusted party, though it has no knowledge that it's being used as such. It depends on the bank generally disclosing the right information in a reliable manner.

If this can't work, how about cash via certified mail? Any financial service that offers public tracking records that can be checked by the recipient. If there is just one service that offers such a thing, this seems possible.