Prowess in battle and ability to lead in battle were very important in medieval and ancient societies, and still are if your country is under threat. There are advantages to having civilian leadership of the military, but with the current government allowing the Thalmor the run of the place, they have forfeited legitimacy anyway. Elisif is a good puppet, I think you mean. You're meant to feel sorry for her, but I don't see any competence, and she certainly doesn't have the respect even of people in Solitude.
Solitude is a rich city, which is not the same thing as well run. The Imperials are certainly gifted at exploiting their provinces.
Ulfric will have his advisors, too, and if he prevails he agrees to submit to the moot so it's not as if he's overturning Skyrim's system of government.
I understand that. Torygg is stated as having "some martial training", although given his age, it is unlikely he has had much opportunity to win his spurs. Given the mentors he would have had access to, it's possible he
could have become a skilled warrior, particularly if Ulfric had decided to work with him. This still fails to address my point: a medieval society where the existent succession can be sidestepped by killing the guy in charge is a recipe for disaster. Not "overthrowing the guy in charge", which was already common both in the real world and in Tamriel, but just "hey let's fight, winner gets crown".
I wouldn't call Elisif competent either. Not yet. But with the advisers she has, her empathy for her subjects, and a little time, I believe she could be a good ruler. In the mean time, she would have Tullius on-side to assist with military matters; it's stated by one character (can't recall who) that one of the reasons Tullius is in Solitude is to advise Elisif. I was never particularly taken by Ulfric's current advisers, who seemed more like old war buddies appointed to various posts, but YMMV.
Last point: Ulfric agrees to the Moot
after replacing any Jarl who would oppose him. That's an overthrow in my books.