Something else entirely. You don't need other galaxies or a universal reference frame. Even if the gyroscope were the only thing in the universe, it would be easy to tell if it were spinning: a very tiny person standing on the inside rim would be held down if it were spinning, or free floating if it were not.
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".
Is what the entire article is about! That our idea of an intertial frame of reference is incomplete.