So this is perhaps speculative to the point of being incoherent, but there is this interesting stuff starting on the middle of page 87 of Rognes' Galois theory monograph (arxiv.org/pdf/math/0502183.pdf) about MU being a "near maximal ramified Galois extension of 𝕊." There is a way of building MU by iterated (Hopf-)Galois extensions, each of which picks up another chromatic layer (Ravenel's X(n) filtration), and the algebra of functions on the Galois group at each level is a graded polynomial algebra on one generator.

The office of the author of this related paper arxiv.org/abs/2008.01706 is down the hall from mine, and they said feel free to send them an email about this question.

This is fantastic. Thank you very much for your comprehensive answer.

Got it, thanks!

@andy I am very confused, that paper seems to be about stabilization of homology of families of groups, or things stabilizing with respect to their decompositions into irreducible S_n representations. I am asking about a system of braid group representations whose decomposition into irreducible braid group representations stabilizes (i.e. becomes limited to induced reps, or some such thing). Can you point me to such a result in that paper?

I guess maybe I was thinking about the action of the fundamental group of the base space of a fibration on the homology of the fiber. Oops.

@andy wow okay I guess I didn't know that the action of a space's fundamental group on its homology was trivial! Hm. So I guess that's a bad example. I'd still like to know if there are any representation stability results for representations of the braid groups.

Agree with David. Consani is a huge part of that project, especially on the algebraic geometry.