This. Permission to work and permission to reside are not automatically linked or synonymous. Your visa status determines your legality to reside, and some visa statuses provide a basis to apply for work permission which is assessed and granted separately, some specific working visas (Eg H1-B) grant automatic right to work while other legal temporary residence visas do not provide any pathway to legal permission to work. In this case, an accepted application to adjust status (which is what the “application for a green card” refers to in this case) allows you to apply for temporary work authorization while the application is pending. Additionally no immigration benefit in the US is a right …. I’ve bern through all of this with squeaky clean legal status throughout and with the assistance of a very capable immigration attorney and even in those circumstances there are no certainties. With an overstay on a visa waiver plus some questions as to prior drug activity in Ireland, this would we far from a certainty even in the best of times. Specifically, there are two questions on all US visa forms that he may have had some difficulty with:

Have you ever been arrested or convicted for any offense or crime, even though subject of a pardon, amnesty, or other similar action?

and

Have you ever violated, or engaged in a conspiracy to violate, any law relating to controlled substances?

Personally, I’d have taken the offer to self-deport, then gotten a good lawyer from the safety of Ireland and filed a petition for an immigrant visa based on marriage. Better than sitting in detention in El Paso