A few thoughts from a non-expert with a bit of background.
First, remember the Hubble constant isn't really a constant. It is an empirical measure, it changes over time, the proportionality it assumes is only established out to a certain distance (a few hundred Mpc), and a lot of assumptions go into any measurement of it, so it doesn't really have a well-defined value in the sense of, say, c.
Second, when error bars are shown for this kind of thing, they typically only incorporate a subset of uncertainties that are sufficiently well-defined to be mathematically analysed. The more complicated (model-dependent, involving lots of measuring equipment...) the process is to calculate something, the more likely it is that the error bars are leaving out a bunch of stuff. Physicists tend to know and understand this, and interpret accordingly, but it gets lost in translation to the wider public.
Thirdly, if there is a 'fundamental flaw' in our understanding, it is important to remember that this is only in relation to a particular regime. None of this is going to overturn everything we know about relativity, quantum mechanics and statistical mechanics as they relate to the 'normal conditions' we experience here on earth, let alone any of the downstream science and technology (chemistry, biology, engineering etc).
Don't get me wrong, there is a real puzzle here and resolving it is likely to involve some fascinating new stuff, but sometimes the reporting of it gets a bit simplistic and overheated.