I meant to ask: If there's only a value cap, that'd mean a miner could trivially blow up the blockchain in size by adding a bunch of near-zero transactions and everyone would accept it as valid.
Not exactly, because the transaction fees policy can be such that if you wish to send a near-zero amount then you have to pay a large fee.
I attempted to give a more complete answer in the previous reply, regarding "third tragedy" paragraph in the paper. The data size cap should be aligned with the amount of compensation that the nodes who secure the network get, otherwise it would be unprofitable for them to provide this security.