If you stick to centralized solution then that would work nice and might be good enough. However Kryptoradio allows to do it by just listening incoming transactions and blocks.
In the pilot we are just forwarding the transactions as they come. But in future we may send extra information if we receive a transaction trying to reuse already used transaction. That will help in the race condition you described: 1. wait for a transaction 2: wait some seconds if there comes an alert about other transactions trying to consume the same inputs.
That might help, but this is an issue which can be addressed after the pilot starts.
We can do it. It's just data. Of course long waves carry very scarce amounts of traffic. DVB-T or even better approach would be using satellite. There are digital satellite technologies like DVB-S which could carry that kind of data. It's only a financial issue.
The idea is that the payment receiver doesn't need to do it. The payee needs to have some kind of uplink (like a mobile phone, amateur radio, or anything).
There are mobile payment applications already, but they are doing it using centralized service and they need make deals with payment agents. With Kryptoradio it is possible to receive payments without any extra middlemen.