Proof of eligibility to work is one's Social Security card, normally. You don't need a birth certificate for that. Now, if you want to make the Social Security card a valid ID for the purposes of voting...
Somehow, the outrage about voter identity fraud - which empirically is so low that reported cases are in double digits across the entire country - is when it's election time, and politicians of a certain bent suddenly find out that poor and non-white people, for some mysterious reason, don't really support to cut top income bracket taxes even further.
Somehow, outrage over this situation only surfaces when it's time to send as much of a low-agency population as possible to the polls.