From the case file, however, it seems like they entered with a J-1 visa and before their authorization to stay expired, they applied for asylum.
Weird but that's how it works. Only after your deportation order has been issued and you do not contest it for an X number of days does it become final and then you have to leave.
If you are still in the country past that date you become an illegal and you can be deported. In the meantime you can basically continue with your life. But you are building on quicksand.
I think this man made the mistake of reasoning that he's useful to society so therefore society would welcome him.
I don't necessarily agree with the immigration laws in the U.S., but I think you have to live within the laws a country has instated.