Um, yea. That's what "public vote count" means.
As for the technical competence of our nation's election administrators, have you not been paying attention? I've met many many. Great at elections. Terrible at computers. Completely and utterly reliant on the vendors. Who've manifestly demonstrated their complete inability to code their way out of a wet paper bag, much less be entrusted with the foundations of our democracy.
Edit: I'm not sure why I'm getting downvoted - am I misinterpreting what specialist is saying? For another example, look at HTTPS - it secures your communication without any need for knowledge of how it's doing so, you just need to know that if you see the green lock icon you're "safe" (though I'm aware of all the usability issues, like how everyone just ignores warning messages when something goes wrong). Is there something fundamentally different about public voting systems in this regard?
The only electronic voting system I'd probably be OK with is one that is totally open-source, software and hardware. The implementation must be completely transparent.
Even then, fraud will happen. Always has, always will.