A claim that I heard once from Saito was that koshi-nage was originally practiced with the nage standing upright - koshi garuma rather than what we think of as koshi nage today. The change (supposedly) was made because the ukemi for koshi garuma was too difficult for most people.
Muddying the water...Anyway, it does make the perpendicular hips seem logical, if you think of it in that context.
FWIW, serendipitously I was watching one of the Shirata Vids that Stan just put up (I've had them since the eighties, but was asked, and agreed, to not publicly distribute them . . . so I didn't.) And it shows sensei doing a series of koshi nage with a bit of a scoop (hardly any compared to most). But I was kind of frustrated, because I've said to my students for years that how he did koshi nage was without any perceptible scoop, much less going perpendicular, and still you would become entirely airborne. There was no perceptible wind up and no perceptible force used, not even the perception of your weight being levered. One moment you would be standing and then you would be landing and there was nothing "in between." Consequently you couldn't resist, there was nothing to resist, you couldn't "prepare" or "anticipate" because there was nothing to prepare for or anticipate. And the landing could be "rude" because the only thing seemingly slowing your fall was the ground, in other words there was no friction or drag.
For me the problem was, due to the nature of the throw, there was little (visual path, sensational path, etc) I could "trace back" to figure out just how he was doing it. So it was "invisible" to stop, and invisible to reproduce.
So I was left "teaching" koshi nage to my students saying, "This isn't it! Then I would show them the outer form and describe what I just described above, and say something like, "Now help me figure out how it works."
Lately the guys have reproduced it a handful of times (I'm probably the worst of the bunch of us.) which is both exciting and maddening (because our performance is spotty at best). Our conclusion? It has everything to do with what we do inside ourselves and virtually nothing to do with timing, positioning, leverage, (Yes, leverage!) or just about any other "normal" things one would look to to explain or solve the puzzle.
The biggest obstacle towards application in antagonistic practice is ourselves. We get excited, or scared, and start "doing stuff" and, POOF! game over. We've gotten better about "policing" our selves though. So practice is productive.
Anyway, I thought since you shared what Saito sensei said, I'd throw this out there FWIW.