Please see https://github.com/bitcoin/bitcoin/pull/3883 which implements this exact scheme. It can solve some kinds of double spends (probably), but others - like ones done by corrupt miners (see bitundo) - can't be solved this way.

Lawrence's motivation for this BIP is essentially to act as a backup in case the Bitcoin native double spending protections end up being too weak to be useful. It reintroduces a notion of centralised trust as a layer on top of the Bitcoin protocol, but only for cases where the seller/recipient feels it'd be useful. In this way it gives us slack: if someone is able to reliably double spend and the merchants losses due to payment fraud go up, we can fall back to TTPs for a while until someone finds a solution for Bitcoin, or we just give up on the Bitcoin experiment, but hey - at least we now have a better intermediary protocol than SWIFT :-)

In practice of course this is something payment processors like Bitpay and Coinbase will think about. Individual cafes etc who are just using mobile wallets won't be able to deal with this complexity: if we can't make native Bitcoin work well enough there, we're most likely to just lose that market or watch it become entirely centralised around a handful of payment processing companies.