Sake is also umami-based, although usually not as intensely umami as dry vermouth.

Usually. There ARE some sakes which are blow-the-top-off-your-head umami. The Funaguchi Kikusui honjozu sake (the one that comes in a can) is wonderful, and is pretty deep into umami. It's also good enough that, if I did a cocktail with it, I'd make it the centerpiece -- I don't THINK I'd pair it with a gin, because I think that'd just get too confusing. But I would happily use a more basic sake in a cocktail like that.