There is a different mathematical justification as well in terms of Bayesian reasoning.
The claim is that “evidence” in Bayesian reasoning naturally acts upon the log-odds, mapping prior log-odds to posterior log-odds additively. To see this, calculate from the definition,
Odds(X | E) := Pr(X | E) / Pr(¬X | E)
= Pr(X ∧ E) / Pr(¬X ∧ E)
= (Pr(E | X) × Pr(X)) / (Pr(E | ¬X) × Pr(¬X))
= LR(E, X) × Odds(X),
Where LR is the usual likelihood ratio.
So when we take the logarithm of both sides, we find that new evidence adds some quantity—the log of the likelihood ratio of the evidence—to our log of prior probability, in this phrasing of Bayes’ theorem.
I sometimes tell people this in a slightly strange language, I say that if we ran into aliens we might find out that they don't believe the things are absolutely true or false, but instead measure their truth or falsity in decibels.
So another perspective on what logistic regression is trying to do, is that it is trying to assume linear log-likelihood-ratio dependence based on the strength of some independent pieces of evidence. You can weakly justify this in all cases, using calculus and assuming everything has a small impact. You can further justify it strongly for any signal where twice as large of a measured regression variable ultimately implies twice as many independent events at a much lower level happening and independently providing their evidences for the regression outcome. So like, I come from a physics background, I am thinking in this case of photon counts in a photomultiplier tube or so: I know that at a lower level, each photon is contributing equally some small little bit of evidence for something, so when I count all the time up together, this is the appropriate framework to use.