proving you didn't do it

Re: At this point, the owner of that router has no way of proving that the email wasn't sent with his knowledge and consent;

The owner of the wifi connection can not prove he did not send the email, and that's often the case: you can't prove you did not do something, but you often can prove that you did. So the onus is on the Police to prove he did, otherwise he must freed. That's the way it works in a 'normal' democratic society.