No. The ideal fix is to go in the opposite direction. You're absolutely right about that part. If any documentation stands in the way, that documentation should be cheap (free if necessitated by the voter's circumstances) and easy to get.
To register in Texas you either have to have a driver's license number, personal ID number, or a Social Security number or you have to check a box saying specifically that none of those have been issued. You must also have a residential address and optionally a separate mailing address. You then must be able to receive your registration card at an address via the postal service. You don't necessarily need to show your ID, but you must have something identifying you as a resident of your voting precinct.
If you don't have a birth certificate, SS card, state ID, driver's license, or a mailing address that accepts mail in your name, then yes you're going to have a hard time voting. Meanwhile, there are over 400,000 people in my city who are in the country illegally. How much easier should we make it for them to vote?