You can verify that the vote was counted, not what the vote was.
You know what the vote was if you witness the ballot marking and the envelope being sealed. That's the reason polling stations and polling booths exist. There, you are the only one who can cast your vote and you are the only one who can see how your ballot was marked. Then you put it in a bin with all the others, anonymizing your ballot, which is sealed and watched by multiple parties to ensure it's counted.
With in-person touchscreen votes, you can likely change the votes in the electronic record in a way that is entirely undetectable. Assuming you can compromise any of those systems, then changing a similar number of electronic votes is much, much easier to do without being detected than changing any sort of vote involving paper ballots, whether in-person or by mail.
That's why paper ballots are better. Human and computer readable for quick tallies and unambiguous recounts. Fully electronic voting is a really bad idea for the reasons you mentioned.
Seriously, the best voting method so far is in a polling place on a scannable paper ballot with open back voting booths and a simple photo ID check. It verifies that the voter is who they say they are, gives ballot access to only that voter while they mark the ballot and creates an auditable paper trail with anonymous ballots.