It doesn't need a blockchain (and predates blockchain), and it has some very nice properties, such as not requiring electricity proportional to how much an attacker might spend, and uncoercibility (as defined in their paper).
I think elections are poorly suited to blockchain for several reasons:
1. There is an authority (the government), so the lack of centralized trust of a blockchain adds nothing.
2. The core of a blockchain is preventing double-spends (double-votes) by picking which version of the truth is correct via some input proof-of-something. Unfortunately, there's more value in subverting an election than securing it, so it seems likely if the US uses $20k of computing power to secure the "vote blockchain", a malicious actor will simply spend $20m of compute power to reverse votes or otherwise stall out democracy. It turns out requiring work to secure something works much less well if it's not money. On the flip side, if we have to spend more for each election in proof-of-work computation than any attacker might spend, we quickly also are losing.
3. Blockchain's cryptography does not provide nice properties like uncoercibility, and those things would have to be baked on top, and yet can instead be built separately more simply (as proved by academic research like the one I linked and the hundreds of papers that cite it).
All 4 of the properties you mentioned are totally unrelated to blockchain technology. You could already get all the above if the government just ran an api that let users digitally sign and publish votes and check votes signed with their key, and that publicly exposes a full audit log. None of that requires blockchain, all it requires is that citizens get a private key (also required if it were a blockchain), don't have multiple, and that the private key isn't publicly associated with them for anonymity.
When you say "Blockchain would help because you could vote on your phone", what you really mean is "digital voting with cryptography to make it secure would be nice". Blockchain is irrelevant if all you want is a mobile app and cryptography that excludes the generals problem.
The actual problem is not a technical one in the first place. Elections are a mess for largely human reasons. Pretending that all we need is some magic technology to fix this problem is being wilfully ignorant.
This is why people are distrustful to voting machines and other techy solutions. Having everything verifiable in public addresses these concerns.
> There is an authority (the government), so the lack of centralized trust of a blockchain adds nothing.
The trust needed is for the government to distribute keys/votes instead of trusting the government to store and count your votes as well.
> a malicious actor will simply spend $20m of compute power to reverse votes or otherwise stall out democracy.
Yes they could. It is however easy to detect.
> so it seems likely if the US uses $20k of computing power to secure the "vote blockchain"
We would of course embed our votes in the much more secure Bitcoin ledger which is much harder to attack and runs without any help from the government (although they could help make it even more secure).