This only applies to asylum seekers who have registered. Visa over stayers or border crossers would have to be detained and have their lack of legal residency status proven before deportation. In other words, habeas corpus would apply.

That's certainly wrong when it comes to border crossers - the SCOTUS case cited upthread was someone who crossed the border, and SCOTUS held that habeas corpus was not available to challenge deportation. I believe the same logic would apply to over-stayers.

That's because the central holding of the ruling is that habeas is available only to challenge detention - not decisions on whether someone should be deported or not. If the detention isn't unlawful, there's no basis for a habeas writ.

Habeas would be available to U.S. citizens being wrongfully detained, of course. In fact, requesting habeas would be available to anyone who was being wrongfully detained. The problem for most folks that are being detained by ICE (even when they're apprehended at their court dates) is that they're not being wrongfully detained, so they'd never get a writ - or even a hearing on a writ, because they wouldn't be able to plead a prima facie case in their petition. If you're present in the country without legal authorization, you can be lawfully detained, with relatively few limitations. Because you're unlawfully present in the country. Very few undocumented folks were detained, because past Administrations elected not to detain them - after all, it's expensive to keep someone detained, and it's much cheaper to just let them do what they will while they're waiting for deportation.