There is no way to both protect the secret ballot and ensure a public vote count. Can't be done. And most systems (proposed, hypothetical, existing) don't do either.
While I agree with everything you just said, I also know there's only two sure-fire ways to increase participation:
#1 Peer pressure
Things like vote.org will help if they close the loop by showing to your peers (bragging rights) that you've voted.
#2 Competitive races
Competitive races require fixes to our gerrymandering and incumbency protection. Maximally competitive districts (totally doable), public financing (will happen sooner or later), proportional voting (tough sell).
Yeah, I defs remember remember reading about a group that sent out a list of everyone in your neighborhood who voted in past elections with an add-on like, "Make sure you're on this list next year", and they got a whole lot of people registered. I also heard Facebook's "I voted" status-maker increased turnout last time around
I know Brennan Center, Pew Trusts, ACLU and others have been working this for a while now.