As for your solution on registration, section 4 of Article I leave the time, places and manner of elections to the states. Hence 50 different election laws with each states election laws written to benefit which ever party is in power. It is like gerrymandering, one party will holler about Texas gerrymandering but will in turn gerrymander the heck out of New York. The other party will holler about gerrymandering in Illinois but gerrymander the heck out of North Carolina. Neither party wants to outlaw gerrymandering only because if they are in power, they can use it for an election advantage.
I think in short, the fear of something, in this case voter fraud is worth more than solving the issue. But neither side would be willing to give up an advantage if they perceive they have one to solve any issue.