I can not speak of "average" or "typical".

However I just want to clarify what I mean when I say the samurai were more bureaucrats than warriors. The reality is that there's no concrete divide between the two in Japan or really in large parts of the world. You were a warrior if you went on the battlefield and fought, and a bureaucrat if you did administrative stuff. A samurai aristocrat doing both was the norm before the peace of the Edo. It's just that in the Edo there was both no wars to fight and to keep the peace, the Shogunate imported (Neo)Confucianism. Traditional Confucianism defines the 士 (from the Chinese Spring and Autumn era), as a gentleman-warrior-aristocrat. This seems to be the definition of the 士 that Japan decided to import.

Note that the warrior nature of the samurai was never actually stripped. The very first code in the "武家諸法度" (bukeshohatto Law Code for Warrior Families) of 1615 says

文武弓馬之道、専可相嗜事

The ways of Civil, Martial, Bow, and Horse. These are things you should dabble in.

In other words, a samurai is in fact supposed to learn martial ways. The bukeshohatto was reissued every few decades, and while eventually bow and horse disappeared from the 1685 issue onwards, civil and martial never did. However in that same code there are a lot of laws against fighting.

So in the Edo period, there was actually a lot of martial arts school that taught a lot of samurai, probably most commonly in kenjyutsu. It's just that, besides running these schools and teaching, martial arts had no real use. You'd have to be exceptionally good to get any kind of advancement through martial-arts.

But, while like I said I can't give you "average" or "typical", from what I can see martial arts schools, especially sword schools, seemed to have been wide-spread. At least some of these sword schools turned into breeding grounds and gathering grounds in the bakumatsu for both sonnō jōi activists and shogunate supporters like the Shinsengumi. A lot of the leading figures among lower class activists in the bakumatsu have their certificates/diploma in one sword school or another. Both Takechi Hanpeita, the head of the sonnō jōi group of Tosa, and Kondō Isami, the head of the Shinsengumi, headed their own martial arts school.

Now it's entirely possible that the arrival of the Perry expedition created a boom in martial arts. However, even if that was the case (and it's very likely it was) we need to also recognize that there was at the very least some sort of widespread popular tradition in the martial arts and schools, as demonstrated in these diplomas, for these men to flock to.