> There is something fundamental in the universe that we don't understand.
Good! I hope there's lots of fundamental things we don't understand well yet. We can then enjoy improving our understanding.
"The paper" below is https://arxiv.org/abs/2007.00082 == https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.12... which is the topic of the fine article linked at the top.
> If this result holds, it could be an amazing leap
I'm not sure if one can use the term result here. In the paper there is just a write-down of a theory that is deliberately designed to match an extra couple lines of astrophysical evidence that previous write-downs incorporating Milgrom's function (see https://arxiv.org/abs/1112.3960 for a partial catalogue of dozens of families of them) did not. Indeed, the paper's abstract says this clearly: "We discuss phenomenological requirements leading to [the theory's] construction and demonstrate its agreement with the observed [CMB & matter]".
The theory in the paper doesn't match all the troublesome-for-relativistic-MOND lines of evidence (choosing to focus on CMB polarization) and the paper's authors are uncertain about several. See e.g. the first paragraph (starting at its third sentence) in the Discussion section on p. 5 and the subsequent paragraph. Note (weak field) quasistatic approximation. They carry on to an admission that they struggle to suppress unlikely imprints on the CMB, and propose to introduce further dark fields to do so. These fields will have their own density distribution and dynamics. The only really nonstandard thing is that they have a strong preference for treating these as gravitational degrees of freedom (modified gravity) rather than generators of stress-energy (matter).
From a theorist's perspective it's neat that they were able to do this, because it's been known for a little more than fifteen years that it's hard to write down such a theory and have it be apparently self-consistent. It is not, however, apparently complete. Indeed, returning to your "there is something fundamental ... we don't understand", the paper says: "Absence of ghosts to quadratic order signifies a healthy theory that could arise as a limit of a more fundamental theory. We do not have such a theory at present".
The ghost condensate they go on to discuss is essentially a non-zero average correction to the values predicted by their theory in the low energy limit. Specifically they appear to be trying to control the non-Gaussianities in the CMB associated with their scalar's dynamics corresponding to an aether (yes, really) having a rest frame that is not the CMB rest frame. (Their unit-timelike vector field A^\mu also breaks invariance under Lorentz boosts, which probably impacts ultra-high-energy cosmic rays if extragalactic, leading to different ghosts.)
> predicts ... an entire class of matter that we've been unable to find any physical proof for other than cosmology (and cosmology is the thing we want to explain!)
This objection shows up often in online forums, but seems to ignore the pattern of discovery of particles that do not participate in electromagnetism.
Roll back to ~1930 where we want to explain the statistics of beta decay and have Pauli postulating a particle which wasn't previously observed in nuclear physics, and defied observation for forty-five years. Or to ~1910 when people were scratching their head over the atomic masses of heavy water and other chemical isotopes leading to the proposal of a proton-massed electromagnetically uncharged particle. Neutrinos (non-relic) are hot dark matter. If they were heavy enough not to zip away from galaxy clusters, they'd be cold WIMPs. If free neutrons didn't decay so quickly, with a suitable distribution (bb relic field?) they could also be candidates for WIMP cold dark matter, and also their discovery by 1932 (i.e. before Bevatron or at least the larger Berkeley deuteron cyclotrons) would seem optimistic.
There may be good reasons to doubt particle cold dark matter, but that it hasn't been detected in the 40 years since Peebles proposed it does not seem like one.
Additionally, particle physicists are looking for a variety of new particles to solve totally non-gravitational problems in the Standard Model. Several of these could be good candidates for particle cold dark matter. Indeed, cold dark matter could turn out to be a mix of several of these. Who knows? It'll be fun finding out.