Just to be clear, it's not fine at all in any "this is a fine physical theory".
A statistically unlikely arrangement of cosmic objects relative to earth only - earth in particular, violates not just the properties of the big bang but "the cosmological principle" and common sense. It's a garbage theory, a headline that makes a math person just have a "how stupid do they expect the average person to be" reaction.
No, I don't think you have this right. There was no physical theory claimed, just a very surprising observation (though of course it turns out to be bogus). If it were true it would require an incredible explanation, but that's not at all the same as just proposing an outlandish theory with no motivation.
If you take the approach you describe, I think you'd have incorrectly dismissed the observation that distant continents look as if they could fit together like jigsaw pieces, as it implied the impossible theory of continental drift. Isn't that just the same sort of outlandish, clearly ridiculous observation as "there's an asymmetry in the way galaxies rotate"? Except of course that one observation is correct, and the other isn't. You can falsify it just by checking the observation, rather than dismissing it out of hand as impossible.
Edit to add: maybe a better comparison would be to the violation of parity conservation in the weak force, as proposed by Yang and Lee, although I just learned from wikipedia that the actual experiment was performed by Wu. If you think that isn't a valid analogy, I'd be really interested to learn why.
TL;DR; all your analogies are less extreme than the claimed observation because they don't violate the principle that physical laws apply uniformly throughout space/time.
Theory or data/observation is a sort of layered thing. The lowest data level is the data that the Webb or whatever telescope gets, then you have the theory that what's being represented in galaxies and you have all suppositions of standard cosmology and in that you have a posited theory that galaxies as viewed from earth rotate in a "given direction".
The thing about this theory/observation/whatever is that it's a pattern that, if true, would only be visible on earth (or in vanishingly small area relative to the claimed area of the pattern). If such a thing were happening, it wouldn't just contradict current physical laws. It would contradict the paradigm science has had since Newton that physical laws apply uniformly throughout the cosmos and especially that the earth isn't the "center" of the universe. The observation of continents fitting together or violations of parity conservation aren't analogous because they involve things that can be meaningfully observed anywhere.
All this is to say "extraordinary claims require extraordinary evidence". As far as investigating. I could or someone could, spend the effort needed to investigate this. But science actually should have one level of claim+evidence that's "interesting, let's investigate" and another that's "oh dear, that's really crank stuff and you'd need to truly vast evidence before I'd even look". Other science would be overwhelm by bullshit.
