Unfortunately thats probably true today, but not the correct solution. Clients SHOULD let you choose where to send notes (in my opinion all notes, but at the very least DMs).

NIP-65 relay lists should probably let users designate an inbox for private messages. This way, your client COULD know where your recipient wants you to send private messages. You could then specify that you only want to receive DMs to nostr.wine, and then all DMs sent to you would be behind AUTH.