I think the only (and easiest) way to fix this is with centralized government identity infrastructure. Right now every agency has to spend some of their budget to verify identity instead of relying on someone whose main job is that.
But I think a lot of the US, myself included, opposes central identity.
We can fix this in decentralized manner, but we won't. Require signed application in front of notary. Lie to notary while presenting fake ID can bring real charges.
There will be a few edge cases (disabled), though after a few prosecutions the fraud will mostly stop.
But I think a lot of the US, myself included, opposes central identity.
Therefore we are in our own self made hole.