Good question!
In Virginia, we don’t seem to have any rule or case law that addresses this question. We do have a rule of civil procedure regarding substitution of parties in the event of a death, but it’s clearly not meant to apply to a criminal proceeding. I strongly suspect that an appeal is mooted by the death of the appellant.
My gut feel is that the underlying conviction should be voided, as well. An appeal is a critical part of the system for deciding guilt or innocence. A defendant who dies before appellate review is completed has not obtained final adjudication of his claim, most especially the chance (as you mention in the OP) to correct any errors made by the trial court. There is no constitutional right to an appeal, but once that right is conferred by statute, it may not be indiscriminately denied.
Looking to federal law for guidance, my “gut feel” reaction seems at least defensible; there is a general acceptance of the principle that the death of a petioner during the pendency of his appeal abates the appeal and the underlying prosecution ab initio. In other words, his “good name” is cleared, since the entire prosecution void from the start - considered as if it never happened.
The basis for this seems to be Durham v. US, 401 US 481 (1971). In that case, the Supremes said that death pending direct review of a criminal conviction abates not only the appeal but also all proceedings had in the prosecution since its inception. The Court noted the unanimity of opinion of the lower federal courts in this regard.
However, in 1976, in Dove v. US, 423 US 325, they dismissed a petition for writ of certiorari after notice that the petitioner had died, and overruled Durham to the extent it was inconsistent with the Court’s ruling in Dove.
With that patchwork of decisions in place, the distinguishing characteristics between them seem to be as follows: Durham stands for the proposition that death of a defendant pending an appeal as of right, or direct review, results in abatement ab initio, and Dove providing the rule that for the death of a defendant pending discretionary or collateral review of a conviction, the conviction stands.
In short, while it depends on the state, I would say you can make a good argument for death during an appeal resulting in the appeal beasing abated, and the underlying conviction being erased.
This was a very interesting question!