If we judge that being Tamil is not enough to warrant asylum in the United States we can just insert him back across the border after taking fingerprints, etc., I assume.

Yes, but the operative terms in the sentence above are the "we" who will "judge" whether being Tamil is not enough to warrant asylum.

The plaintiff argued that since he was physically present in the United States, he was entitled to a hearing on whether he qualified for asylum. That the "we" couldn't just be ICE or Border Patrol or an LEO, but it had to be a judge (whether administrative or Article III) who would afford him an opportunity, governed by due process, to submit evidence on whether his situation warranted asylum.

And the Court said, "Nope." Congress has provided that you can be deported without a hearing, and the Constitution doesn't require one.

Immigration advocates want Thuraissigiam to be read as limited to border encounters, but the basis of the opinion doesn't support that. The Court held that Habeas Corpus offers judicial review (and relief) from unlawful detention - from being imprisoned or held in custody. It doesn't grant a right to judicial review of your asylum claims. IOW, you can assert that the government doesn't have a legal basis for detaining you, but the Suspension Clause doesn't mean they have to give you a hearing on whether or not you can be deported. That analysis isn't limited to the border.

Under Thuraissigiam a person who is detained as unlawfully present in the country could seek a habeus corpus writ challenging their detention....but like the petitioner in that case, that is an empty effort. If you're present without authorization in the country, the government has the right to detain you. The detention isn't unlawful. You never had the right to enter the country, and you never acquired the right to be at liberty within the country. As noted in the opinion, even if you are paroled into the county and live for years in the U.S., you are still legally treated as though you were an applicant for admission at the border - and until such admission is legally granted, you can be detained indefinitely.