If you place the ID after the ballot is handed out (by a printer that is also used to fill in the ballot). Then this systems still doesn't allow proving of votes.
The ID here is meant to identify a ballot, not a voter. It should probably be something like a UUID. The aim of this system is to allow cross-checking between the scanner and the physical ballots.
I got that, but you can still kind of prove it. Your know your ID + your-vote. This is likely the only valid ID+vote combination you can know before results are counted. That's when I'd "ask" you and late verify it.
If you want to verify the machine is working, just put the ballot in the standard bin and add those IDs in the counting phase. That seems fine in principle and make it easy to check the tech is working as intended. You'd end up with having list of all individual votes available, maybe even to the public. I'd be worried about people throwing statistical algorithms at that. You better also find a near perfect method to randomize order...
The ID here is meant to identify a ballot, not a voter. It should probably be something like a UUID. The aim of this system is to allow cross-checking between the scanner and the physical ballots.