The CMB was (and is) observed a lot: https://en.wikipedia.org/wiki/List_of_cosmic_microwave_backg...

The CMB is the evidence for a hot big bang (note, this is not the same as a singularity) although of course observational cosmologists want to trace back from the first starlight to the CMB and have various plans to do so. Surprises in that "dark age" are more than welcome because they are bound to reveal things about low-density momentum-energy.

> assume dark matter and dark energy are real

Nobody has put together a self-consistent physical cosmology model that reasonably matches actual observations of the shape, scatter, and shift of galaxies across our sky, without the theory containing some additional nonluminous tension and pressure terms that take effect earlier than the surface of last scattering and persist to the present day. The self-consistency problems are expected to blow apart consistency with observation. (Details in e.g. Famey & McGaugh 2011 §9).

The need for a tension term to match observations arose even in 1917 [1], and the more detailed 1929 Hubble observations supported the idea of a constant tension independent of visible sources; that is what modern observations still support. The need for an additional pressure term became obvious from the 1960s Rubin observations; the term was pretty simple initially and that simplicity was well-supported through the 1980s and 1990s. One could through that period support theories wherein the pressure term appeared everywhere luminous sources appeared in some straightforward function (e.g. Milgrom 1983). Modern observations have several advantages in the observable details, and those support a rather more complicated interplay between luminous sources and the pressure term, particularly at scales much larger than kiloparsecs.

If you can find a theoretical mechanism that produces the observed small anisotropies of the CMB without introducing some tension and pressure acting on an almost entirely homogeneous photon gas with an almost perfect (albeit highly redshifted) blackbody spectrum, well, that'd be amazing. It should of course also be compatible with the actual observations of less-highly-redshifted molecular gas spectra (and the Lyman-alpha forest), and how those lense around rather than through radio-clumpy foreground galaxy clusters.

There are a lot of galaxies in the sky, so many that we've discovered that their shapes are pretty regular across an enormous redshift-range. That's additional observational evidence for the spatial flatness of the universe. Even fairly gentle spatial curvature would distort the shapes of face-on spirals at high redshifts, and that is not what we see.

Even the Hubble Ultra Deep Field image imposes pretty significant constraints. It's littered with galaxies and with goodies like the Gravitational Lens Galaxy Cluster CL0024+1654, which have been imaged in various wavelengths by both ground and space telescopes.

The match between the small anisotropies in the CMB and galaxies at increasing redshift strongly ties the hands of speculative theorists. Those are the people who do the "modelling and simulation". They are stuck having to match actual observation, assuming their goal is to theorize about the single example we inhabit. Bolshoi and MultiDark had to match Sloan, not the other way around.

> bad type of modelling

Whose word are you taking for this?

> we only have a single example. In contrast to ... weather .. where we can keep ... comparing to the real world

As said at the top, we've kept comparing the scatter and shape of galaxies and the scatter and shape of fluctuations in the CMB. How is dealing with evidence questioning the standard value of the H_0 parameter not "refining the model"?

Also, today we only have a single example of an Earthlike planet's weather data. And until the late 1960s at the earliest we had weather data of any sort for only one planet. So was the numerical weather modelling work done in the 1950s (e.g. by Charney, von Neumann et al. on ENIAC) also "an especially bad type of modelling"?

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[1] E Schrödinger, “Über ein Lösungssystem der allgemein kovarianten Gravitationsgleichungen”, Physikalische Zeitschrift,19, 20-22 (1918) cf. https://arxiv.org/abs/1211.6338