Randomness extractors from Bourgain’s breakthrough- applications?
...would it help in any way with the issues discussed here?


No, not really. There's always been a gap between myriad theoretical mathematical extractor constructs and those used in commercial validated TRNGs.  I honestly don't know why as that seems like an unlikely dichotomy, yet it demonstrably exists. The two should ideally converge to a common output entropy rate, but they don't :-(

Most creditable TRGNs are based on min. entropy extraction via the Left Over Hash Lemma: $  \epsilon = 2^{-(sn-k)/2}   $ where $s=H_{\infty}$per bit input to the extractor. This lemma is unaffected by Bourgain.

So no.