Well, you could take some kind of unique voter ID number and run it through a one-way hash. Then you could check your vote, verify that it was accurately recorded, and view how each vote was tallied. The current system is terrible and better solutions exist.
The current manual system works. Why is it terrible? Most results are available within 24-48 hours an electronic system may speed that up but for no appreciative benefit.
What's to stop you from selling your hashed voter ID to a vote buyer, before the tally is published?
He verifies that your hash corresponds to the vote he purchased, and pays you the money (or doesn't beat you up, fire you, etc. I'm using "sell" in the generic coercion sense)
For a paper ballot, it's much more difficult and risky to prove delivery, yet you are fairly certain it's been counted.
The grandparent proposes a perfect counting guarantee, at the cost of providing every voter and coercer a crypto proof of the ballot value, a very bad compromise.
It's easy to configure the voting booth to provide privacy for the voting itself, while making it risky to photograph the vote. You don't know the vote was actually delivered, ex. he might have gotten a fresh voting paper or punched/stamped excessive options to annul the vote. There is also an electronic evidence trail of the fraud - any caught vote seller will immediately cooperate with the authorities and rat on the ringmasters.
Real world security works on "just hard enough not to be worthwhile", not absolute guarantees.
Plenty of places make it illegal to take a photo in the voting booth (or polling station generally), and the voting booth is visible enough to election officials (albeit with your back to them and hence the ballot not directly visible) that it'd be very hard to do without arousing suspicion.
electronic devices and photographic equipment aren't even allowed in my local polling places. i had to leave a kindle outside. that is as it should be.
