You're just assuming this level of organization that applies to SUPERHERO TEAMS. Like everyone takes a vote, or there's some big meeting to come to a decision that everyone must abide by. There's NO superhero team in WWII, NO membership roster, NO rules that everyone must abide by, NO headquarters or equipment, NO communicators. You can't have a "Civil War" if there's no "Union" to begin with. Whatever loose level of familiarity they had with each other is rooted in those Golden Age cover images where you see three or four characters from the same publisher working together in some kind of wartime scene, more along the lines of a Brave and the Bold or a Marvel Team-Up (and following up on that in the modern day, that's exactly what we see the most in Project Superpowers - loose groupings of 3 or 4 characters working together). Where PSP is lacking in detail, I rely on the actual Golden Age comics as the primary resource for information about these characters. Where it doesn't directly contradict anything from those Golden Age comics, I am not making any assumptions beyond what has been shown (with the caveat that I will allow that the Golden Age covers that don't depict actual stories inside the comics can be taken as some sort of "untold tale"). Admittedly there are a lot of blank spots, but there's nothing stated in PSP that ever implies any sort of permanent team structure for WWII superheroes. If you're inferring that, you're basing it only by analogy to Roy Thomas' retro-superhero comics, something that has no basis in what's actually shown or stated in PSP, and further probably projecting some sort of leadership role to Fighting Yank analogous to Captain America. Since PSP isn't giving us a complete summary of the WWII history of these characters, only a couple of highlights that involved a lot of them together, I think you may be filling in a LOT of the blank spots by analogy to The Invaders and All Star Squadron where that analogy is unwarranted... your primary backup information source should be the Golden Age comics that featured these characters.
Previous to the Japan mission, you see them together in one scene where they've been recruited for this mission to reclaim Pandora's Box from Hitler, and the CO explains to them what's going on. Later, Fighting Yank gets them together to try to convince them - unsuccessfully, so clearly these characters don't all see eye-to-eye about everything - to sacrifice themselves to be trapped in the urn. (Here we have another example where the heroes disagree with each other on the proper course of action -- yet, no heated arguments ensue, no punches are thrown.) So, based on that, you assume this group of characters works exactly like some sort of un-named All Star Squadron? But even in the scenes mentioned, there are many characters missing (seen later in the series, or in Alex Ross' Golden Age character gallery). Those two scenes are atypical (again with reference to the Golden Age stories, where they rarely teamed up, except on covers); lacking any name or assemblage by official request into a permanent team with an origin story -- there IS no superhero team. These scenes are more akin to a Marvel or DC "crossover event". But I do allow that this is where the title Project Superpowers (never mentioned in the story itself) appears to mislead readers. Where you see the word "superpowers" used in the series to refer to these characters, you should be thinking "superheroes" or "metahumans", not "All Star Squadron".
To the degree that they all felt patriotic, they lent their abilities to the war department, but most of these guys were not full-time soldiers. They had civilian IDs that they maintained throughout the war. When Fighting Yank says "we" went, he's talking about the other superheroes he knew best, and those he agreed with or got along with best. He's speaking for himself, he's not the spokesman for every person who had superpowers in WWII. The only way they'd have gotten together as many supers as appeared in Japan is by putting out word through some newspaper, or perhaps a radio speech or something like that. It's hardly an "argument" if you're simply recruiting volunteers who happen to agree with your POV. It's not even clear if Fighting Yank was the one to initiate the idea, or he's simply recounting that he was part of it. Frankly, I'm kind of baffled as to why you feel it's absolutely necessary to project the inclusion of every single character not depicted in that scene as actually being there, though not shown. Why does it matter if they weren't? If they weren't there, it doesn't imply that they argued or fought against the idea, it could be nobody asked them, they were busy elsewhere or couldn't be contacted, just didn't feel it was their job and declined to go, or whatever. But the most logical reason would simply be they didn't agree with the plan, so they didn't volunteer. It does seem a little more realistic and consistent with the depiction of the PSP characters as more than two-dimensional, who have their own individual outlooks and personalities. But it really doesn't matter WHY they weren't there. Maybe some were that you didn't see, who knows. But it just doesn't seem realistic that Every American Superhero That Existed During WWII could, or would want to, be there. It's easier to just read what's on the page.