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.
Your analogy is analogous, and that's exactly the same problem as with the blockchain! Unless you're maintaining your own Bitcoin full node, your integrity comes from the provenance: "my lightweight client trusts this full node not to lie to me". This is the same as your "one guy to sample the papers".
All you need to do is grab a copy of the day's paper from your local convenience store, and compare your results with the Times website and two randomly-selected peers (selected from a distribution carefully chosen to ensure that each day's graph is connected). Any discrepancy will be obvious, and undeniable (since you have the physical artefact as a certificate of duplicity), so anyone who discovers a discrepancy can blow the whistle. If no whistle is blown, then either there was no discrepancy, or there is a big conspiracy (i.e., one large enough that blockchain wouldn't have saved you either).
The problem is not all that difficult. The main advantage of Bitcoin is that it's a good enough solution that many people don't feel the need to think about the problem any more – even though it's a marginal improvement over the prior art, with major downsides of its own.