What you have just described is an ExpressVote voting system, manufactured by ES&S (https://www.essvote.com/products/expressvote-3/). Here is an example how-to on using it from Micigan (https://www.youtube.com/watch?v=ebqktli8bRk). The only salient difference between what you describe and the actual system is that the paper run through the machine is also audited (to guard against someone ballot-stuffing by creating additional ballots when nobody is looking).

If you want this, the next step would be to get involved at your county or state level (depending on how your state makes voting technology decisions).