Would it be correct to say that the reason why judges do not have power to set aside a jury verdict of acquittal in a criminal case is precisely that that would violate a right to a trial by jury? Would there be other reasons besides that?
Criminal conviction by a judge after acquittal by a jury would violate the Double Jeopardy Clause of the 5th Amendment. A criminal conviction involves both a finding of fact and the application of the law, and a judge (unless this is a bench trial) doesn't find facts, he makes judgements of law. Overturning an acquittal after a jury trial would thus be in essence a whole new trial (moreover one where the defendant was not given a renewed opportunity to defend himself).
It's a "separation of powers."The judge's main job is to interpret the law (and the intersection of law and fact). Unless both sides agree to a "bench" trial, which is to say that the judge is then charged with trying facts, as well.
The whole purpose of "trial by jury" was to create a fact finder independent of the judge. Which is why either side can (often) demand a jury. A judge can overturn a jury finding, but only on grounds of "law." If a judge were able to overturn a jury finding of fact, the "separation" would be moot.