IMO, if Gabriel is still alive, she knew ahead of time where she was going and had a plan. I do not think it would be that easy to find an underground adoption couple in a weeks time. If someone has Gabriel, they are hiding him and know it is illegal.
She did not give him away randomly IMO.
I am sure there are people who are desperate for a baby and would hide him, but you would not meet them in a park.
If it were not for the Smiths involvement I would have little hope of him being alive.
In other words, I don't believe anybody can just go anywhere, give a baby away and it disappears off the planet.
But Eliz was good online, so maybe she did hook up with the right couple, but the odds to me are very slim.