While Shinto like that may not have been all that politically active as a single power, Japanese mythology and cult practices related to the Kami were to my understanding still important in that period to the Japanese population. If my mind isn't wrong there were for example both shrines constructed to the Kami by samurai warlords.

Not I wouldn't advocate Shinto as a seperate religion beside Buddhism and Christianity. However due to the syncretic nature and fusion, to my understanding and I could of course be wrong, between Buddhism and the practices regarding the Kami I think that building shrines devoted to these and gaining traits regarding their worship would be nice.