(I assume it’s not that simple, but it’s very hard for me to understand and reason about data that involves looking back in time and space simultaneously)
Way back when, we used to joke about "Murphy's Variable Constant", the fiddle factor required to make any equation balance nicely. It's sad that the Hubble Constant is now a variable.
Seriously? How can it be fully-determined by things we can't detect, let alone measure?
There are separate constraints on almost all of those densities, from different observations. For example, big bang nucleosynthesis puts constraints on the density of baryonic matter, dark matter and radiation, while the CMB essentially tells us the density of radiation.
Cosmology is based on a web of different observations, tied together by well understood physics. It's very difficult to fudge things to match one observation without messing up the results for other observations.
It turns out that roughly 68% of the universe is dark energy. Dark matter makes up about 27%. The rest - everything on Earth, everything ever observed with all of our instruments, all normal matter - adds up to less than 5% of the universe. [1]
That doesn't quite live up to my idea of "well understood physics".
[1] https://science.nasa.gov/astrophysics/focus-areas/what-is-da...
Here's cosmology in a nutshell for you:
You take the commonly accepted theory of gravity (precision-tested on solar system scale [1]) and assume that it works unmodified on cosmological scales, impose the "cosmological principle" (actually a maximally simplifying assumption rather than a bona fide principle) and get the FRW model [3].
The instability of FRW's static solution, the observed redshift of distant galaxies and the cosmic microwave background then lead you to the big bang, and applying (actually well-established!) nuclear physics to primordial nucleosynthesis results in good agreement with observed element abundances [4] - apart from lithium, of course [5].
But never mind that, you have bigger fish to fry, like solving the horizon [6] and flatness [7] problems which you inadvertently introduced along with the big bang.
Happily, it occurs to you (following Guth and Starobinsky) that an early period of exponential expansion would help there, and also explain the apparent absence of exotic relics from GUT symmetry breaking. GUTs are of course anything but well-established physics [8], but if high energy physicists can dream, then why can't you? Thus, inflation is born [9]. Some naysayers may claim that you have no idea what supposedly drove it, but you know better: you actually have way too many ideas. There are actually more inflationary models than any sensible person could possibly keep track of, and no way to tell which - if any - of them is correct. That's by design. Known physics can't do the job, so they must be based on hypothetical new physics capable of escaping detection in every physics experiment ever performed to date.
But never mind that, because once again, you have a more pressing problem to solve: even granting that all assumptions you've stacked up so far are correct, your model doesn't fit the data. There's just not nearly enough matter in the universe to reproduce the observed flatness, expansion history and element abundance which you just tied up so brilliantly with inflation. Then again, if you are willing to assume an undetected (and quite possibly undetectable) inflaton, why not some far more mundane, pretty much ordinary matter which just happens to be undetected (and quite possibly undetectable) too? There are even boring astronomical reasons to take the notion seriously. Crank up the density of this unknown dark matter [10] to 85% (!) of all mass in the universe, and your model works - kind of.
See, after you stuffed all that matter into your model, some enterprising astronomers went out and did their best to measure the rate at which it's causing universal expansion to slow down. Surprise: they obtain a far better fit to the data by adding a freak term to your model, Einstein's old cosmological constant [11] and making it so large (about 70% of all energy in the universe!) that it dwarfs your dark matter. Which means that according to the once-again patched model [12], the expansion is not slowing down; it's accelerating.
Oh well, you say, if I can have exponential expansion in the early universe, why not now? A flood of dark energy models follows, again more than any sensible person could possibly keep track of, again based on anything-but well established or even verifiable physics, for the same old reason: known physics can't do the job, so it must be something which eluded every physics experiment ever performed to date.
But that's still not the end of the story. Because as noted in the article linked at the top of this page, even this latest iteration produces inconsistent results when fit to two different measurements of the same quantity. So to the extent that there are "a lot of well understood physical processes that produce clear predictions about what we should observe", they are saying that the model is, in fact, wrong. Again.
[1] https://en.wikipedia.org/wiki/Tests_of_general_relativity
[2] https://en.wikipedia.org/wiki/Cosmological_principle
[3] https://en.wikipedia.org/wiki/Friedmann%E2%80%93Lema%C3%AEtr...
[4] https://en.wikipedia.org/wiki/Big_Bang_nucleosynthesis
[5] https://en.wikipedia.org/wiki/Cosmological_lithium_problem
[6] https://en.wikipedia.org/wiki/Flatness_problem
[7] https://en.wikipedia.org/wiki/Horizon_problem
[8] https://en.wikipedia.org/wiki/Grand_Unified_Theory
[9] https://en.wikipedia.org/wiki/Inflation_(cosmology)
[10] https://en.wikipedia.org/wiki/Dark_matter
Besides, we can measure these densities just fine.
There are very few assumptions in the cold dark matter paradigm (you can describe them all with just a few parameters). Meanwhile, there are many different observational probes of dark matter, and they all have to agree with the predictions of the theory.
There are many ways that one could seek to falsify the cold dark matter paradigm. The thing is, so far, the paradigm has passed all the tests people can think of.
This is already generally being accounted for (assuming we have the correct cosmological model for the Universe). The CMB measurement is probing the Universe at an early state (when the Universe was ~380,000 years old) and the Hubble "constant", H(t=380,000 yr), was different at that point from what it is now, H_0 = H(t=13.7 billion years). The comparison to the current Hubble constant (e.g., determined from Cepheids) is made by evolving the CMB Hubble "constant" value to today, under the assumption of our best cosmological model model.
Of course, if we have the wrong cosmological model, then that could be a reason for the discrepancy, and that might point to "new physics". Alternately there may be some systematic in the Cepheid technique that is causing our H_0 estimate to be somewhat off.