So there are minute differences between theory and observation.
That is why there are people working on better theories.
However, whatever new theory that explains the anomalies won't significantly change predictions, and will likely contain many components of GR.
In the mean time, GR does the best at explaining motions in the sky.
It, along with particle physics, explains how stars are powered, and how brightly they will shine.
It explains the variations in measured g at various points on the earth.
It explains why clocks run at different speeds at different altitudes.
It explains why starlight gets slightly bent when passing massive objects.
it explains the cavendish experiment.
(Not that you will believe this one because it involves spacecraft, but Gravity Probe B directly observed earths gravity well and gravitomagnetic field.)
So yes its imperfect, but it is very accurate, and new theories can't toss out the notion of gravity (something that pulls masses together, some thing that causes masses to change the nature of space-time around them, something that keeps the planets in orbit, etc...). The effect is independent of the ways we try to quantify it, and will retain properties that the new theory will still have to be able to predict.
When you come up with FE physics more accurate than GR, tell me. Because that would be really cool.
Or at least FE physics that had some math. You have to start somewhere.
