I don't know about other voting machines, but the ones here in Northwest Arkansas had a receipt of your vote behind plexiglass that you could watch print out as

I don't know about other voting machines, but the ones here in Northwest Arkansas had a receipt of your vote behind plexiglass that you could watch print out as you made your selections. Pretty similar to what you suggest, except that I don't believe they use the paper for anything unless their is worry of fraud.