For what I can tell, after the clickbaits and the sensationalisms, there is an actual unexplained discrepancy in the values of the Hubble constant measured with different methods. This means that something is not right in one of the methods (perhaps both), or that we are missing something else.
And yes, the Hubble constant is closely related to the age of the universe, so it’s basically the same issue.
This is actually a good thing, because the “crisis” can be the first clue to discovering some new exciting science! Just look at how the “ultraviolet catastrophe” kickstarted the field of quantum mechanics a century ago
OK, I’ve actually watched the video this time (the last part, as you suggested). It seems that the author decided to leave out the debate about the Hubble constant and talk about other phenomena in the early universe, which also have their uncertainties.
As far as I can tell, the Hubble tension is still a thing, and JWST only confirmed the measurements of the Hubble telescope. See for example this recent article on ars technica.