The simplest solution is to have the electronic device print out a copy of the voter's balloting. The voter can look it over to see if it's correct, then deposit it in a sealed box (the same way punch cards are currently deposited). If no one challenges the electronic vote totals, the paper backups are ignored. But if there is any doubt about the integrity of the electronic numbers, the paper is available to be recounted.
This does not reveal a voter's identity. What it does do is make fraud much less likely to even be attempted, since the paper backups would have to be altered to match the electronic results.
I'm an electronic engineer, and I have absolutely no confidence in the security of purely electronic voting systems. There are all kinds of ways they can fail or be compromised, and there is no way to recover in the absence of a paper backup.
I've also personally been involved in very close elections and one recount (our candidate won a Park District race by 3 votes out of 95 thousand cast), so I know how crucial it is to be able to verify the results.
Don't kid yourself, both the Republicans and Democrats have been guilty of election fraud at different times and places. We must not presume that either party is inherently trustworthy.
If you want more sophisticated schemes for secure voting, while maintaining anonymity, those are also available. See David Chaum's white paper on Secret-Ballot Receipts and Transparent Integrity.
On the other hand, the tried and true methods used to corrupt all the other systems, most particularly the PAPER ballot systems, are still quite useable and well known.
This guy in California is just trying to preserve Democratic party dominance ~ nothing more to it than that. If you think any Democrat is concerned with the integrity of any voting system, you are confused.