"The court's opinion was authored by Chief Justice Salmon Chase, himself a former cabinet member under Abraham Lincoln and leading figure in the Union government during the American Civil War."
So, that doesn't mean he was incapable of ruling on the legal issues in the case. Many, if not most, judges are former prosecutors, yet they hear criminal cases all the time. Chief Justice Taney backed Andrew Jackson to the hilt while serving as his attorney general, yet did an about face as Chief Justice. Attorneys, when representing clients, are advocates for their client's position. When attorneys become judges they are supposed to be neutral, listen to the arguments of both parties and apply the law; if they can't do that, they should recuse themselves - it has worked that way for over 200 years.
Regarding Chase & Texas v. White, I don't see anything indicating that either party to the suit asked Chase to recuse himself. Remember, although Chase wrote the opinion, four other justices reached the same conclusion. Even if he had recused himself, it still would have been a 4-3 majority in support of the ruling. Interestingly, I didn't hear Davis, Stephens or any other southern partisans complaining of Chase's bias because of his ruling AGAINST Lincoln's military tribunals in Ex Parte Milligan, 71 U.S. 2 (1866). But as I noted above, a judge is only biased when he rules against your position.