Proof of citizenship is not the same as the driver's licenses people are issued by their state.

Not everyone has ready access to proof of citizenship in order to register to vote. It gets even more difficult if your current legal name doesn't match your birth name, e.g. if you took your husband's name.

Not every eligible voter has or needs a government issued ID. For example, retired people who don't drive. For them to get to the DMV to get an ID just to vote would be a challenge.

The US has large rural areas where government offices are hours away.

All of this adds up to significant barriers to eligible voters. There's a reason even the GOP isn't bending over backward to pass the SAVE Act.