The mistake which resulted in the big bang and dark energy is actually very simple.
That it was never fixed in the course of 100 years, I feel justified to call cosmology foolish.
The mistake is just the selection of units!!!
I wrote a book to explain the many mistakes in cosmology, and how to fix them.
Perhaps this post removes the incentive to read the book.
A red shift is a measured change in a wavelength. This change is essentially a ratio between observed and original. It is described with a dimensionless value called z. z is actually the ratio of the object's velocity compared to the velocity of light, c.
If you multiply z times c, the result in km/s.
In the 1920's, astronomers were measuring red shifts of galaxies. Soon a pattern was observed. The red shift increased as the distance increased. Cepheids were extensively measured in the 1920's.
When one is found, the Cepheid enabled the distance to be calculated for its galaxy.
The initial Hubble's law would have used this ratio, z / Mpc, for its Hubble's Constant.
Vesto Slipher is given the credit, or blame, for treating that z as a velocity.
This mistake changed the ratio to this: km/s / Mpc.
My book spends a few pages justifying this z value for a galaxy comes from the hydrogen atoms in the IGM.
This ratio could be called the "Inter-Galactic Medium factor" or IMF, or whatever (the book has another), but its units must be z / Mpc
The updated Hubble's Law distance equation becomes:
D = z / IMF, where D is in Mpc. and z is the galaxy's red shift.
When every galaxy gets a z value then none gets a velocity. Each should get an IMF.
For those galaxies with no Cepheid they could be assigned their IMF from a nearby galaxy which has a Cepheid.
There is no justification to expect this IMF is uniform across the entire universe. That is the expectation for Hubble's Constant. Even its name says it is a constant. Unfortunately, cosmology is in crisis mode when its value changes however it is measured.
That also results in the uniform expansion of the universe.
IMF cannot be treated as a constant.
In my personal opinion, this is embarrassing for cosmology.
If someone in the last century had just returned this ratio to using z not a velocity, then there would be no big bang and no dark energy.
I call this foolish.