There are also the many examples of relativity being falsified, which are open problems. For example the flyby anomaly.
Ah, yes. There are tiny, but apparently real, differences between theoretical velocities and actual velocities, measured in parts per billion. That means we aren't accounting for
everything that affects near-earth flybys. We don't, for instance, know all of the details about slight variations in the earth's gravity field because we don't know all of the details about the exact distribution of mass inside the earth. Because of this, very small perturbations to orbits compared with what is expected, of sufficient magnitude, are possible. Of greater immediate economic significance, because of this imperfect knowledge of the earth's interior, prospecting for mineral deposits is not easy.
We would have to know a lot more about this and other imperfectly known physical conditions before a we could say this "falsifies relativity" with any confidence whatsoever.
Without a model of gravity, a great many round earth arguments start to fall to dust.
Even if this were true, it's moot because there's a model for the effects of gravity which works splendidly on the scale of the solar system. But it's not true, so even that is moot.
"But..." you (and at least one other very long-winded poster here) may splutter, "you can't explain exactly what causes mass to exert an attractive force other mass!" This is true, but we do know mass has that effect on other mass in great, even if incomplete, detail.
Believing that
not knowing everything is the same as
not knowing anything is equivalent to believing that not knowing one's bank balance perfectly, to the exact cent, is the same as having no money at all. Both are preposterous.
You mention "many examples of relativity being falsified" and give one which doesn't do so. Do you actually have
any?