There would have to be a public hash-to-transaction mapping somewhere.

Yes. There is a database of utxo - unspent transaction outputs. To be correct - it is an index of blockchain file.

For a user running a full node, this is basically what happens. On initial installation, they download (from other peers on the network) a copy of the complete block chain, containing every transaction ever executed. So when a new transaction arrives, they have the data to trace all the transactions that led to it.

To make this more efficient, the client maintains a database of valid unspent transaction outputs (utxo), indexed by transaction hash (or txid). When a new transaction T arrives, the client checks that the inputs of T all correspond to distinct utxo's in the database. If they do, and all the signatures are in order, and everything else about the transaction makes sense, then the transaction is accepted. The utxo's corresponding to the inputs of T are removed from the database (since they are now spent), and new utxo entries are created corresponding to the outputs of T.

For "thin" nodes, which do not keep all this data, they have to request it from other network nodes which do. This complicates matters somewhat and requires a certain amount of trust in the nodes providing the information.

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.