Use client `send_private_note` method

Currently the faucet sends private notes via the note transport by instantiating a `GrpcNoteTransportClient`. Instead, it could just use the client integration with the existing `send_private_note` method.

Also, now that the client in Send + Sync, there may be room for changes in the faucet architecture by sharing the client behind a mutex.

Context in https://github.com/0xMiden/miden-faucet/pull/191#discussion_r2738733647