But men do get mugged and murdered, including especially when they're out alone.

So why do you chalk this up to "yay freedom for men" instead of "boo we don't care if men die / get mugged"?

If we dip back into the 1980s, when the patriarchy was stronger by pretty much every measure, we find that even then men were disproportionately the victims of robbery: https://uat.bjs.ojp.gov/content/pub/pdf/rv.pdf

(That could well be because men were out more, but even so, it very clearly demonstrates that men were in fact in danger.)

You make a lot of good points, but this one seems backwards. This seems primarily an expression of callousness towards men.

The "men run most societies" argument as you presented it is also rather weak, because in the places you talk about there are exactly zero formal barriers to the participation of women (and you quote women who started in politics when the informal barriers were far stronger!), and while you identify informal impediments, you don't show any evidence that they are strong enough to account for the low numbers; and furthermore, you don't make an argument that it even matters in a representative democracy who happens to actually be in the legislature, because the point of democracies is supposed to be that the needs of the population is reflected. Now, as far as I'm aware, it happens to be the case that the needs of women aren't (thus far) as attended to by men in politics, but leaving support for this out of your argument does no favors for its persuasiveness.