The Big Bang didn't happen: What do the James Webb images show?

No, it didn't. Particular supersymmetric extensions to the Standard Model made testable predictions of what particles the LHC should detect at different energy levels, none of which have panned out.

String theorists like to take credit for those predictions (or at least they did until the predictions became notorious for their wrongness) because they claim that string theory is what justifies those particular supersymmetric extensions to the Standard Model. That doesn't mean we should let them get away with it.

The first, in fact, is not even finalized yet, because there are still unresolved issues about the actual value of the Hubble constant: different sets of observations point at different values and the discrepancy is not fully understood. (Google "Hubble Tension" for much more.) But this whole general line of research is not about "ruling out" a theory, and it's certainly not about putting some better theory in its place; it's about refining our existing theoretical model by adding more and more reasonably confirmed detail about the actual distribution of matter and energy in the universe and its actual expansion history.

The second is a particular kind of hypothetical particle that has never been observed, still never having been observed. That doesn't "rule out" the existence of that kind of particle with 100% certainty. It just means we still haven't observed it. (Even the article you reference notes that there are still axion models that have not been ruled out.) And it doesn't change our best current model of the universe at all.

The third "rules out" a class of models that nobody has ever used. Similarly, the fourth "rules out" a model that was not even built to be used, but was expected to be ruled out, at least at the level of accuracy of these particular measurements. (That accuracy, as I understand it, is still well short of the Planck scale, which is where most physicists expect any "quantization" of spacetime itself to show up.) These results are useful in that they narrow the space of possible models that scientists might consider trying to use in the future; but they bear no resemblance whatever to what Eric Lerner claims (incorrectly, in my view) is shown by the James Webb telescope observations, which he claims "rule out" the entire Big Bang model.

If we don't have a better one, and if our current model makes reasonably good predictions about many things, even if its predictions aren't as good about some things, we wouldn't say our current model is wrong because it isn't wrong. What we would do is to start having more conversations about our current model's domain of validity.

Perhaps some historical observations will help. Consider the history of physics, or more specifically mechanics. Before Galileo and Newton the best model of mechanics we had was Aristotelian physics. When Galileo and Newton came along and gave us a better model, we dropped Aristotelian physics, and now if you asked most physicists they would probably say Aristotelian physics was wrong, period, without feeling the need to add qualifiers or explanations. This is the kind of case I think most people are thinking of when they talk about falsifying a theory with contradictory data.

But since that transition, there have been no theories of mechanics that have been falsified by contradictory data. When special relativity came along, we didn't drop Newtonian mechanics, and physicists now don't say Newtonian mechanics is wrong. They just say it has a limited domain of validity: it's an approximation to relativity that works reasonably well for relative speeds much smaller than the speed of light.

Similarly, when general relativity came along, we didn't drop Newtonian gravity and say it was wrong; we just clarified that the domain of validity of Newtonian mechanics is also limited to weak gravitational fields. And even if GR hadn't been discovered, physicists would still have continued using Newtonian mechanics even though there was data known in the 19th century, like the extra precession of Mercury's perihelion, that was not consistent with Newtonian mechanics. Newtonian mechanics still gets taught to physics students today, and is still used for practical computations in all kinds of contexts.

When Eric Lerner claims that "the Big Bang didn't happen", he is claiming that the Big Bang theory is destined to go the way of Aristotelian physics. But none of the data he cites justifies that claim. At best the data justifies a claim that the standard Big Bang model that assumes homogeneity and isotropy is an approximation with a limited domain of validity--which cosmologists all already agree on anyway! What we are still working out are the exact limits of its domain of validity. In other words, cosmologists in general expect our current Big Bang theory to work like Newtonian mechanics: we might discover models in the future that are more comprehensive and to which our current model is an approximation in some limited domain, but that's very, very different from it being falsified.