I'd always kind of imagined the reactionary geometers were defending an order in which their tools were imperfect finite approximations that yielded insights into perfect infinite truths, where the original sin of the revolutionary analysts was in saying that "yes, and with compactness and continuity, many of these problems have their α-and-ω in finite descriptions".

Is that a fair take? Would it be one, even if it were ahistorical?

I like that view, but I believe the conflict was more about "old vs. new". Geometry was very old by that point, ancient, and it carried a lot of personal gravitas by being associated with Euclid, Archimedes, Thales etc. (Galenic theory of humors enjoyed similar ancient intellectual prestige, hence its long and bitter retreat from the scene at approximately the same time.) It was also "obviously right", in the sense of "everyone can look and see for themselves". Even uneducated peple can verify that a certain line touches both circles etc. No wonder it was an attractive safe haven for conservative minds.

Meanwhile, analysis was not yet particularly rigorous and it took several decades to converge on a standard apparatus and notation that could at least be understood coherently by other mathematicians. (Laymen tend to struggle with it until today.) Add the political dimensions of being seen friendly to the French into the mix, well...