But that doesn't distinguish between the gyroscope spinning and the existence of a sufficiently massive object in the appropriate place, right?
General Relativity may seem counterintuitive, but experimentally it really does seem to be the way our universe works. Starting out from the position that absolute motion is meaningless has proven remarkably fruitful. It's also something a lot of very smart people have been thinking about for a hundred years. Whether or not you find Mach's arguments plausible, they're fundamental philosphical claims and can't simply be dismissed through a half-baked thought experiment.
In that case, it wouldn't be drawing you outward no matter where you were located. So, spinning would be "you feel a gravitational force", and a ring of mass would be "you don't feel a gravitational force".