
Not really, no, because current cosmological models are not derived solely by looking at cosmological phenomena and coming up with phenomenological equations that describe them. (That is not to say that past cosmological models didn't do that; only that our current ones, roughly since the early to mid-20th century, don't.) They are derived by starting from laws of physics that already work in other domains, and seeing what those laws say about the universe as a whole.
> to be absolutely clear what the assumptions of the model are, even the most trusted ones
Cosmologists are clear about that.
> maybe this is just an issue in the popular press?
I think it is, since what I see in actual textbooks [1] and peer-reviewed papers is not at all like what you are describing.
[1] A good reasonably current textbook is Liddle's Introduction to Modern Cosmology.
For example, here's a quote from the book "An Introduction to Modern Cosmology" by Andrew Little (third edition, published in 2015, page 70):
It is just about possible given present observations that this matter can be entirely baryonic, since this is marginally consistent with Equation (9.3). However, many models based on low-mass stars and/or brown dwarfs have been excluded adn it is probably difficult to make up all of the halo with them. A popular alternative is to suggest that this density is in some new form of matter, which is non-baryonic and only interacts extremely weakly with conventional matter.
It looks like you are entirely convinced that dark matter exists beyond any reasonable doubt. Good for you. In the end, this doesn't affect your life or my life all that much.