When you say "An unlocked wallet has valid keys, but no one can send MRT to it, you cannot receive MRT to it, and you cannot receive invite tokens to it either.", do you mean a "LOCKED" wallet?
If you cannot send to an unlocked wallet, then who can you send to?
When you download the wallet or create a wallet on the web, it generates a valid address, but that address is not 'beaconed' so you cannot receive Merit (MRT), and since the balance is 0, you cannot send any either. You need to first receive an invite token (INV) and it will beacon and unlock the wallet, at which point you can receive funds and send as you wish.