That's not the only argument the majority made in Herrera. Habeas corpus is a specifically protected right by the Constitution. The argument wasn't just "this is impractical." The argument was "a guilty conviction that sentences a person to death is not a violation of habeas corpus even if you are actually innocent and we all agree that they are innocent".
Here's a good article about it. Scalia's claim was in the dissent, so this isn't established legal doctrine in the US, and the Supreme Court has in fact ruled in favor of the convicted on more than one occasion, but as the article shows it has actually happened as practiced legal doctrine in at least one case.
Reading that it's specific to an 8th amendment claim.
It definitely is horrific. It looks like further Supreme Court cases allowed actual innocence claims under other standards, or for specific types of evidence. All of these were close decisions though, and I fear what a more 'conservative' court will allow.
> Although this is a correct ruling, the Court's opinion is not likely to significantly protect innocent people from being executed. First, the Court failed to recognize key points of distinction between the evidence of innocence that Schlup presented and the evidence of innocence presented in Herrera v. Collins. Because Schlup's evidence was much stronger, the Court should not have ruled on the issue of whether the Constitution bars the execution of a factually innocent person. The implication of the Court's silence is that full habeas hearings are unavailable on straight-forward constitutional claims of actual innocence. Second, the Court's analysis of the fundamental miscarriage of justice exception does not appear to contradict the argument that the exception is a rule of permission. Thus, after Schlup, Federal courts will likely be free to dismiss the habeas petitions of State prisoners even where new evidence of innocence makes it "more likely than not that no reasonable juror would have convicted."
