How it ever made it so securely into the paradigm of cosmology, astrophysics, astronomy and physics, I do not understand.
Because when it comes to fudges, everything else is more fudge-like than dark matter.
See, we already have reason to believe there are sterile neutrinos. All observed fermions except neutrinos come in both left- and right-chirality, while observed neutrinos only come in left-chirality. Further, observed neutrinos have been proven to have mass, and even before they were proven to have masses the usual quantum mechanics equations implied that if they did, then counterpart right-chirality neutrinos would also exist. And these counterpart neutrinos would, per the quantum mechanics, fail to interact with the strong, weak, or electromagnetic forces (thus why they were called "sterile"), but they would have masses and interact with gravity.
So, then we've got a whole bunch of observations of the universe that imply there's a whole bunch of mass out there that doesn't interact with the strong, weak, or electromagnetic forces (and thus is "dark"), but does interact with gravity. These include not just the usual galactic rotation curves, but things like the Bullet Cluster, which does gravitational lensing not as if it has two separated centers of mass on its wings (which its visible matter most definitely is), but one central concentration of mass (which would have to be non-visible).
Then there were the observations of galaxies whose apparent rotation curves were much slower, relative to their visible mass, than most others. That doesn't make any sense at all if rotation curves are determined by visible matter; you have to invent new patches to any non-dark-matter explanation of galactic rotation curves to explain these cases, while with dark matter you say, "Collisions between galaxies can just by random chance change their dark-to-visible matter ratios; if dark matter is true, we'd expect to see some low-dark-matter galaxies."
So, "sterile neutrinos exist and make up much of the mass of the universe" explains a bunch of observations both at cosmic scales and quantum scales. A computer model of tidal forces just plain doesn't; it "solves" the usual galactic rotation problem without explaining the unusual cases, or the Bullet Cluster, or anything else.
Similarly, there's also the axions, which solve the strong CP problem in quantum chromodynamics, and also could have mass while not interacting with the electromagnetic force and having minimal interactions with the strong and weak forces. Like sterile neutrinos, they too can contribute to "dark matter" astronomical observations while also solving a quantum mechanical issue, instead of explaining away only one "dark matter" issue.
In short, many major "dark matter" candidates explain whole slews of astronomical observations as a side effect of very simple extrapolations of standard quantum mechanics, while all the non-"dark matter" theories are fix-only-one-issue fudges invented post-hoc to correct one class of observations.
That's how the consensus that there's dark matter became so fixed. Everything else requires multiple unrelated theories to explain as much as (e.g.) "sterile neutrinos" does all at once.