Don't get me wrong - I'm completely against backdoors. But when you shift the argument into "we won't do it because it's impossible", you're already agreeing that it should be done, while your argument won't hold because it is in fact possible.
There's three kinds of people: 1. Non-technical people (theresea may) that want backdoors are don't care about whether it's possible or not. 2. Technical users, with a vague general knowledge of cryptography, and the imprinted thumb rule of "backdoors are bad" 3. People with actual knowledge in cryptography which had already been doing research about why it is possible for years. Just a teaser: https://en.wikipedia.org/wiki/Kleptography
Of course, the real issue would be the scale and the distribution of access to the backdoor to various agencies.