It isn't surprising to me because the standard of journalism seems to be one of "debunking". In this approach you start with something that you don't like, in this case Mond, something you do like, usually the status quo, and then throw a bunch of pasta at the wall in the hope that something will stick with the reader. There is usually little to no attempt at an honest review, only a biased statement salad. It's pretty depressing, but I guess it sells ads and doesn't bother the establishment.

Can someone ELI15? Does this work claim that because galaxies fly around each other, potentially around some bigger center of mass, rather than in straight lines, their rotation curves look differently/are actually different, and nobody checked before?

Disclaimer: the authors are good friends of mine and I'm a computer scientist, not a physicist, so I'll try my best to explain it:

The basic idea here, is that the reference frames used to measure galaxy rotation curves may not be inertial. that is, they might be influenced by rotational or other accelerative forces due to the galaxies' motion in the universe. Traditionally, it's assumed these frames are inertial, meaning they are free from such influences, which simplifies calculations but might overlook significant effects, and that's the key point that has been missed and is addressed by the paper, showing that you can't overlook those effects.

if we consider these non-inertial effects akin to acknowledging that galaxies might not be moving in straightforward, stable paths but could be "flying around" each other or a common center, then we can account for the observed rotation curves differently. They suggest that the forces arising from these complex motions could mimic the effects attributed to dark matter.

AFAIK, this has not been rigorously checked in the past. However, there are several other phenomenon that are explained by dark matter / dark energy besides this discrepency, so people in the field are not very eager to give up on dark matter just yet

Sounds like you are confirming my TL;DR; explanation.

This paper is very weird. They basically assume we are rotating with respect to the center of a galaxy and that is why the see different rotational velocity compared to when we would not be in a rotating frame. This could be the case for a single galaxy. But how can we be in a rotating frame towards the center of all galaxies? That doesn't make sense to me.

So the math would be correct for a single galaxy, but this cannot happen for all galaxies at the same time.

What is interesting though, that they basically find a model with a single additional parameter per galaxy (angular velocity omega) that fits most of the rotation curves very well. However the most simple explanation how we get this omega is not by assuming fictitious forces from rotation frames, but simply real forces from dark matter.

So, actually the paper isn't saying that we are all spinning in sync with the center of every galaxy out there. What it's really getting at is that each galaxy might be spinning or moving in its own special way because of the specific gravitational forces acting on it locally.