How are you "your own bank" without actually having the banking software?? A private key does not a bank make! I can generate all sorts of keys and sign all sorts of messages; that doesn't mean I have money! The "most important feature" of bitcoin is the block chain: the full history of transactions and coin bases. Without this, your signed messages and private keys are for naught.
If they can transmit transactions with a light client it's equal to having the banking software to me.Maybe I got it wrong but I know that nobody can spend your money without having the private key. A private key kept private makes it impossible for anyone else to spend your money.
You do know that your logic is faulty. You do not have a web server at your home to download or use stuff from the Internet or from your mobile banking system. This is stupid and utopic and it will never happen!