Scientific models are always being refined to fit new evidence. Some of the headlines have been really click-baity and inaccurate and some of the models have already been adjusted to account for the early galaxy formation. There is a difference between “this is surprising and not what we would have predicted” and “this destroys our theory.” The data raised some questions that the current cosmology models can’t account for. But that doesn’t mean they are totally “wrong,” it means they are incomplete. There have been similar problems with our models and dark matter and dark energy. It drives more research and eventually when we understand more, we can make better, more complete models.
Nothing I have seen has implied that scientists are questioning whether the universe had a beginning. They are adjusting their ideas about timelines and mechanisms of galaxy evolution after the singularity.
The Senior Project Manager at James Webb, John Mather is a Nobel Laureate who is an expert in the Big Bang. He is not throwing out the theory any time soon and much of the click-bait “theory in crisis” stuff probably comes from people with a reductionistic understanding of the Big Bang model anyway.