You don't, if you're a philosopher. But the rule is beyond reasonable doubt, not beyond all conceivable doubt.
The reason to disbelieve the "elaborate framing" hypothesis is straightforward: It's really easy to prove, and yet it has not been proved. All you have to do to prove it is find Nina. And yet she has not been found.
In fiction you can make a character disappear by saying "they moved to Russia/Argentina/Outer Mongolia and disappeared". In the real world it really is harder than that. Russia is not on the moon. You can phone up some Russians right now if you want and have a chat. Even if the Russian government won't cooperate with an investigation, there must be private investigators in Russia. All Hans and his legal team have to do is hire a P.I. to stake out the kids and produce some evidence that they're being visited by a mystery woman who might be Nina and suddenly his case becomes a lot stronger. He could still do this now, from prison, if he wanted. Might really shorten that 25 year sentence.
I suppose you could argue that Nina is so sneaky and so well connected that she and her kids have truly vanished from mortal ken, like Osama bin Laden. I haven't followed the case, so I don't know if Reiser tried to argue that. If he did, it obviously didn't work.
And I suppose you could argue that the woman hates Hans so much that she's willing to be separated from her kids for 30 years (or forever, via suicide -- there's a Sherlock Holmes plot for you) just to maximize his suffering. Good luck selling that "reasonable" theory.
The jury system is intended to insure that no innocent man is ever convicted...
Um, no. That jury system would be called "never sentence anyone". This system aspires to spare the innocent, but there is never a guarantee. And I have to say that, in the annals of potential injustice, this is a pretty weak example. Literally thousands of innocent-but-convicted Americans would be free today if they had even half of the legal resources that Reiser hired and then apparently chose to ignore.