Mercury's motion does violate Newtonian gravity. General Relativity however simplifies to Newtonian gravity for various low-dimension applications though (i.e. an Apple falling from a tree, the orbit of the moon for most astronomical guidance purposes over that distance etc.)
In much the same way we don't account for the curvature of the Earth when building a house's foundations.
That's my point though: when you violate a fundamental physical law, it can't just be wrong. You have to somehow also account for why it doesn't happen all the time elsewhere (i.e. the effect is small to within experimental error over common measurement cases) and by necessity the new theory to explain it has to encompass the predictions of the old i.e. the benefit of Newtonian gravity was that it easily recovers the behaviour of nested epicycles of orbital motion (and made new predictions successfully where epicycles could not).
I still don't see how this is possible. In GR gravity travels at the speed of light, in Newtonian mechanics it is instantaneous. I have been unable to find an acceptable explanation for how these two very different universes can be compatible.
NASA also does the same thing, so don't feel bad.
The orbital motion of Mercury "violates" Newtonian gravity. There are other examples but I'm not trying to make a history lesson. I just want to point out, that science moves forward by impartial investigation of EVERYTHING. It's misguided to assume we know all and that there cannot be some thing we don't understand.
I'm as sceptical as anyone else about the EM Drive, but I want to understand "what the heck is going on with this thing that appears to move" more than I want to dismiss it as "not possible and obviously flawed because $ABC"
There are plenty of possible ways the drive could operate, and if it does work, it has the potential to significantly advance several fields of physics in a way no less valuable than other large expensive "hope this works" experimental physics experiments that no one is complaining about.