It's important to think about the exact procedure you want to use for verifying something. Running with your thought experiment, let's say we publish "the root hash of the whole world" (not too far off from what Keybase did) each day in the Times. Now I open my phone to read some messages from Billy Bob, and my phone needs to get that hash somehow. This is just a thought experiment, so let's say for the sake of argument that it tells me to walk down to the convenience store, buy a copy of the day's paper, and scan a QR code on page 12. The problem with that arrangement (even in thought experiment land, where I'm happy to perform these steps every day) is that all the evil server needs to do to trick me is to put a doctored copy of the Times in that
one newspaper stand. That's not the level of security we were hoping for. To get real security here, I'd need to do some sort of
random sampling of newspaper stands distributed across the country, to build confidence that whatever QR code I'm seeing is the same one that everyone else is seeing. And the kicker is,
everyone has to do this. We can't just pay one guy to sample the papers every day and tell us what the QR code was, because now our security depends on trusting that one guy, and the whole point of peer-to-peer security is avoiding that kind of centralized trust.
I think this is actually a great way to talk about the difficulty of the problem that Bitcoin solved, and why so many nerds were so interested in the whitepaper, long before all the real money got involved.