For self-deportation to happen in significant numbers there has to be a simultaneous three-prong course of action: 1) eliminate the dole (free medical, education, etc.), 2) heavily fine illegal-hiring employers, and 3) repeal the anchor baby laws. Anything short of that would be ineffective.
"2) heavily fine illegal-hiring employers"
And use that money to fund efforts to keep out/deport more illegals.
It is O.K. with me for the anchor babys and one parent to remain - under current law the "anchor baby" is a citizen - as long as the other parent returns to the "home country" and gets in line like everyone else.
It will take no more time, and no less time, than anyone else who is presently overseas and seeking to immigrate.
Meanwhile, the other parent here will remain illegal, until the first returning parent's legal immigration is secured and they can apply to sponsor the parent who remained.