Even if every voter gets a hash and can check that their vote is in the log, you still have a bunch of places where a central actor can misbehave: Deciding who gets to write to the log in the first place, rate-limiting or dropping submissions, or running split-view logs in the event that there's not a ton of replication - hoping that wouldn't be the case in an election.
With a (properly designed) blockchain, you at least push those assumptions into a consensus layer with many writers/validators and game-theory penalties for rewriting its history. It's still not magic; but for something like elections, I'd rather minimize the points where a single operator can tilt the playing field, which is why I was thinking "blockchain" instead of "centralized transparency log"
Everybody gets a copy of a verifiable hash etc when voting, allowing voters to mathematically check their vote.
The kind of knowledge allowing to design such clever algorithms is the real meaning of the word "crypto" (cryptography).