The legislation is crafted so same sex couples can not file the contract together (you have to be "legally able to marry" under Alabama state law). It has nothing to do with "getting government out of marriage" and is just an end run around the federal courts (and potentially the SCotUS) saying that Alabama counties had to offer marriage licenses to same sex couples. If Alabama honestly wanted "government out of marriage" then they would allow any two consenting adults to file the contract: heterosexual, homosexual, mother and son, brother and sister, whatever. They don't want that, they just want a loophole to keep the gays from marrying.
It's not "getting government out of marriage", it's just homophobia with a ridiculous veneer of libertarianism. Should the SCotUS vote in favor of SSM, someone will just sue Alabama on Equal Protection grounds and almost certainly win.