My state uses a system that I think is the best of most worlds. You check in at one table, they give you a "receipt" that you give to another table who trades that for a little card. The card isn't linked to you at all and it gives you access to a voting machine. You enter your choices into the machine, which prints out a paper ballot that you can review and verify before putting it into the box as you leave. So, your vote is recorded by the voting machine for quick tallying, but your vote is also recorded on paper that is human and machine readable for verifiable re-counts if needed.
the opportunity to recount is not itself an adequate protection for an online counting machine. in my county for example, recounts are only triggered for close elections. they do zero auditing for normal wins.
The system needs to be secure in the primary case, not only in the audit case.
Now if you're saying the counting machine is offline, so that it's verified during the normal voting process, that would be more acceptable.
