It's not actually clear that there is any problem with current theory - it may just be that our understanding and application of it is incorrect. You can have a perfect mathematical model, but if just one of your premises is incorrect, the output can be nonsensical.

I am a physicist (well, I studied as one, before going technology entrepreneur), and have always sat in the "dark matter is bullshit" camp - so my opinion is likely tainted.

I think a far more likely explanation, as this paper could suggest, is that something far more mundane is happening.

For instance, we already understand tidal forces - that bodies can become tidally locked, that there is an inertial frame shear that drags bodies along around a rotating centre of mass, and exchanges energy between an orbiting body and the centre.... So why is it infeasible that galaxies are tidally locked, and the lack of fall-off in radial velocity is down to the gravitational shear from the galactic core being far more intense than we anticipate, causing the behaviour we see. It doesn't require a rewrite of relativity, "just" of galactic evolution.

Either that, or galaxies are sprites, and lazily rendered ones, at that.