Honestly, branching the cold war narrative out is probably the more obvious way to streamline the show. It can't be easy to juggle two plot lines, past and present, and have them reflect upon each other without becoming contrived and gimmicky.
But the late '50s part of the show feels more vital than the rest of the show, maybe for its heightened Indiana Jones feel. The bigger challenge IMO is to make the contemporary human characters seem relevant. Especially the twenty-something cast is given mostly filler to work with, against a backdrop of pretty dire kaiju fare.
Although I think that the best Godzilla movies have focused on human drama, I can genuinely understand if viewers of corresponding ages don't identify with Cate, Kentaro, and Corah. They're the biggest albatross around the show's neck, even given the Rodan appearances.