I doubt a tourbillon has better accuracy than your Rolex (it's a chronometer, right?) even though the design is supposed to reduce the effects of positional variances. Perhaps,
@Archer can chime in.
As already noted, the tourbillon was intended to counter the effects of gravity in pocket watches, so in a wrist watch it's a bit redundant since your wrist already moves the axis of the balance around to cancel out errors.
Regarding the Gyrotourbillon, it has been quite as successful product for JLC. It is of course not making a huge impact on the overall watch market, because not many people could afford one. It's quite a complex thing, and speaking to the 2 watchmakers who do the assembly of all of these watches at the factory (this is done in a separate area in the small farmhouse) the accuracy is quite good - I can't recall specific numbers though. Here's a video I shot through a microscope there:
However when you look at the accuracy that can be achieved using other technologies, like silicon balance springs that Omega is using, the tourbillon becomes even less "worthy" from a pure timekeeping standpoint. But of course that's not the point of a tourbillon in a modern wrist watch. At one time you could not purchase a tourbillon wrist watch for less than six figures, and there was some "uproar" in the industry when JLC dared to release the Master Tourbillon that was "only" $40k. I can tell you other companies were not pleased that someone had proven that these could be made in such an inexpensive manner that you could charge so little - it endangered their cash cows.
Of course if you really want one, you can get a Chinese made version for about $1000 - at least you could the last time I looked at these. I recently had a service request for an Android watch that was a tourbillon, and although I've worked on one of these Chinese tourbillon movements previously, for me to perform a service would be more than the watch is worth. Really the only practical option (since parts can't be had) is to replace the whole movement.
But as some have pointed out, mechanical watches are an anachronism...from a purely technical standpoint the world moved on decades ago. Making performance improvements on a mechanical watch is a bit like perfecting the buggy whip...sure there is a market, but it isn't what it used to be. If you look at this whole industry (one I realize I make my living through) it's all a bit strange to be honest...
Cheers, Al