Registrations of individual registrants can be challenged and then a more detailed examination is done. Both Republican and Democrat local or state parties keep an eye on registrations and challenge any the think are bogus.
Extensive statistical checks after elections by numerous statisticians (and by most major political parties) shows that this system is working fine, so there is no justification for rushing a transition to a more rigorous system. There is time to design a system that will work for all citizens.
We could get a long way there by just grandfathering in everyone who voted in the last 10 years, and making sure there is extensive help going forward for anyone who cannot afford the cost of getting the necessary documents to get the new ID or who cannot get those documents.