Has any government ever investigated the rate of false positives in a jury system by faking trials? ... Would this be a worthwhile thing to do?

It's probably worthwhile, but the purpose of a jury was not to improve accuracy. After all, you could just have one or more judges, who would have more experience and thus be less prone to error, determine guilt instead of a jury. The purpose of a jury was to make corruption more difficult (harder to secretly bribe a dozen randomly selected citizens), and to provide a way for the people to negate laws they don't like (can't punish juror for declaring someone innocent).

So even if there were a more accurate way to produce verdicts, there are other factors to consider.