Really, "honest services" is even more vague than mail fraud or wire fraud. With those two, at least the prosecution has to prove you used the mail or the phone.  With honest services, it isn't even that difficult.

What the appeals court will consider is whether there was enough evidence presented to support the verdict rendered.  This is called, in lawyer shorthand, "the weight of the evidence".  The weighing of whether there was enough evidence is done in the light of precedent, i.e., asking whether a similar quantum of proof had previously sustained a verdict.

In the civil context, for a party to preserve an appeal where that point of the argument is titled "The verdict was against the weight of the evidence and should be reversed", the party must have made motions in the trial court at the end of the plaintiff's case and at the end of the defendant's case for judgment as a matter of law.  It roughly goes like this:

[Opening arguments, Plaintiff's case]
Plaintiff's counsel:  "The plaintiff rests [, reserving the right to seek to introduce rebuttal]."
Judge:  "motions?"
Defendant's counsel:  "Your honor, at this point, the defense moves for judgment as a matter of law.  Based upon the evidence and testimony presentend, no reasonable jury could find for the plaintiff."  [the second sentence is the legal standard for the motions]
Judge:  "thank you.  I'll reserve."  (or deny the motion)(see below)
[Defendant's case]
Defendant's counsel:  "The defense rests."
Judge:  "motions?"
Plaintiff's counsel:  "Your honor, at this point, the plaintiff moves for judgment as a matter of law.  Based upon the evidence and testimony presentend, no reasonable jury could find for the defendant."  [the second sentence is the legal standard for the motions]
Defendant's counsel:  "Your honor, the defense renews its motion for judgment as a matter of law.  Based upon the evidence and testimony presentend, no reasonable jury could find for the plaintiff."
Judge:  "thank you.  I'll reserve."  (or deny the motion(s))
[Jury instructions, summations, instructions, deliberations, verdict, entry of judgment]
Losing party files a motion for a new trial, based upon the verdict being against the weight of the evidence.
Motion denied.
Appealing party files an appeal, raising as a point of the argument that "The verdict was against the weight of the evidence."  Another point can be "I should get a new trial because the trial court's denial of my motion for a new trial was erroneous and should be reversed."

I'm far from a criminal procedure maven, but as I understand it, the basic structure described above also holds for criminal trials.

Now, the point (severalfold) is this:  by going after the weight of the evidence in deciding the appeal - "your proof is beyond thin" said one judge - and attacking the government's theory of the case - "I'm not sure what your theory is" was one judge's question - what the appeals court judges did was preclude a retrial and make sure the case was over.  Why?

In tossing the case on the grounds they did, the appeals judges accepted the facts as they were, and did not address any question of law - whether the arguments were allowable, whether the jury instructions were correct, whether the indictment was correctly drawn, etc.  Nor did they decide whether the defendant received a fair trial.  Addressing a question of law or whether the defendant received a fair trial would have required reversing the decision, and sending the case back for a new trial.  Finding that the verdict was against the weight of the evidence would, in the civil context, also likely result in a new trial, particularly if some evidence favoring the loser below had been excluded.  But, since the Double Jeopardy clause protects defendants against the government trying again, there is no retrial.  Simply, no reasonable jury should have found guilt beyond a reasonable doubt.  And, because of that, no more trial.

It would be like if someone was indicted for Crime A because he wore red pants in public.  The state could put on a dozen eyewitnesses to say they saw that person wearing red pants in public.  They could introduce a videotape of the person standing in public, wearing red pants.  They could have a confession from the person, admitting that he wore red pants.  And the judge could instruct the jury that, if they find beyond a reasonable doubt that the person wore red pants in public, then they should convict the person of Crime A.  And then the jury, with their mountain of evidence, would convict.  But when the appellate court looks at the case, they can say, "Wearing red pants in public does not constitute a crime.  You can't be convicted of Crime A becuase you wore red pants in public."

(2) get on with their lives;

(3) make a nuisance out of themselves advocating against the prosecutor/cop who done them wrong (which usually has bad results for the former defendant if done directly), or turn into an activist.

Some states have laws allowing a person to recover some amount of money (often, X dollars per year of confinement) in the case of a wrongful conviction, but the feds (IIRC) do not.  Since this was a federal conviction, no luck.