Here is the analysis of the Ohio appellate court, quoted from the 6th Circuit decision:
Chambers shared some similarity to Sinkfield's case because the defendants in both cases presented the defense that another person committed the crime for which they were charged. The court noted that [the caller's] statement inculpating himself and [the third party] met the first element of Chambers because it was made soon after the murder. However, the court never unequivocally stated the results of its analysis of the second Chambers factor. While there was evidence corroborating [the third party]'s involvement in the crime, the court found it difficult to overturn the trial court's decision to discredit the testimony because it was in the best position to determine the credibility of that testimony. Finally with respect to the third Chambers factor, the court noted that unlike Chambers, where the statement was wholly self-inculpatory, [the caller]'s statement inculpated himself and another party, [the third party].
In holding that the petitioner failed to prove that the state court's decision was an unreasonable application of clearly established Federal law, the court emphasized that Chambers has been limited to its facts, and where a third party was involved, as in Sinkfield, the exclusion of the evidence is not an unreasonable decision.
Its important to note that the petitioner in Sinkfield is alleging actual innocence. He maintains that he didn't do it, that it was the caller and the third party. I'm not entirely sure whether or not the evidence that was excluded would satisfy any of the gateways laid out by the Supreme Court. After all, the trial court apparently let in testimony that the caller admitted to killing the victim, and the jury still convicted.
However, if the evidence at trial showed that two people committed the crime, and the jury simply believed that Sinkfield was the second person, then perhaps Sinkfield gets closer to being heard en banc or a cert grant. The case was heard before three judges that were all appointed by Republican presidents (2 GWB 1 GHWB) and that seems to mean an automatic denial of habeas relief in the 6th Circuit. And yes, the reverse is true as well for democratic nominees, which is just as unfortunate.