The person sending the coin would have to include the whole ownership chain, which would start increasing more and more in size
This alone wouldn't be sufficient, because it doesn't tell you whether the chain of ownership was double spent, or whether those transactions were confirmed in valid blocks, or even if the original coins were mined properly and exist in the longest chain. To know all that, you need to see the whole blockchain, or to trust someone else who has seen the whole blockchain.
The paper explains that the basis of the blockchain is that everyone is aware of every transaction that ever happened, and the order in which they happened. So by having access to the blockchain, you can validate a new payment to make sure it is genuine and not double spent.
Bitcoin nodes are optimised so that they can just validate a chain once and then store a set of its unspent transactions, so it means they don't have to re-check each transaction all the way back to the genesis block. This is just an optimisation though, so that's probably why it was left out of the paper.
You can also use an SPV wallet (see the section in the paper titled "Simplified Payment Verification") which eliminates the need to see the full chain, at the cost of outsourcing trust to the miners.