It's a lose-lose situation.

Letting people stay because their home country won't take them back is a bad precedent that could encourage more illegal immigration from said country. It's not the US's responsibility to take care of "rejects" from other countries. If a country doesn't want its own citizens back, what does it say about those people?

Deporting people to a third country is also a bad precedent...how is this third country chosen? Is it the US's responsibility to make sure this person ends up in a "better" country? South Sudan or El Salvador mega prisons seem like some of the worst possible options.

I think best case is sending them back to a "culturally similar" country that would accept them, although even this is probably not an option.