People would understand that voting online wouldn't be anonymous but I'm willing to bet its a trade that many many people would be willing to make because it's already how absentee voting works.
Please study some history on elections before making dangerous suggestions.
Paper ballots have been used quite successfully for about a thousand years. It's not broken, so why fix it?
We need everybody to trust election outcome.
The thing that I think would be great about using your phone to vote is that more people would participate in the U.S., where voting is optional. Would make no difference here in Australia where voting is mandatory.
NB in an election of a government there is no central trust store. And everyone (including your grandad and thick cousin) need to understand the process and be able to verify it.