Paper ballots work fine and have amazingly low error rates, leave it be.
FYI, the voting in blockchain can be anonymised.
The issue I have with electronic voting of any kind is that it's a huge and valuable target, it's pretty much guaranteed that multiple hostile nation states will want to poke holes into these systems. Introducing crypto doesn't do squat to prevent all sorts of nasty problems, including information leaks and stolen private keys. Given what we know about how most people handle basic passwords, I think paper is probably the most secure form of voting we've ever seen.
The political voting is a complex beast so I am not going to comment on it yet. My main focus is for the voting process like American Idol, choosing moderators in forum.
When you think through this, every voting system has a weak link in terms of verifying who is a valid voter. So you must have a centralized authority here. Thus a blockchain is useless.
This is the hypothetical solution. Let's say there are 1000 voters. You premine 1000 coins then send it to each voter. Voter then can vote with 1 coin. So you can make sure the voting process is valid, and you can hide the sender in the smart contract. Of course, the technicality is not this simple. But you get the idea.
Let's say you record voting based on SMS for American Idol show in MySQL database. Singer A got 1000 votes, singer B got 700 votes. But singer B is more viral and popular in social media. You think if singer B goes to the next level, it will increase the rating of the show much more compared to singer A does. So you decide to bump up the singer B's voting number to 1100.
Blockchain can prevent this malicious behaviour.