I'm absolutely devouring all your hazbin analysis, it's so good.
Would you be willing to tackle the subject of why Vox seems to think acting in a more aggressive, stereotypically more masculine fashion is what will impress Alastor/what Alastor wants him to do? We see Vox is normally pretty confidently, if somewhat subtley, genderqueer in the way he carries himself day to day, whether it be in public terrorizing Charlie at the hotel or when he's reprimanding his most homophobic reporter, and Alastor himself isn't, at least physically, some bastion of masculinity himself (although he does act rather toxically masculine at times, and far more so than Vox, I'd argue, since his actions are internally motivated vs Vox, whose actions are driven by Alastor and not his natural state).
Vox is shown to tailor his approach based off of what the other person likes (or is perceived to like, see his disastrous maraca approach to Camilla), so Alastor's own actions and reactions have to be what lead Vox to think he respects traits like authoritarianism, selfishness, etc.
If there's a subject I'm not willing to tackle, I haven't found it yet, lol.
The short answer is that Vox thinks that's how Alastor wants him to act because it is how Alastor wants him to act. Being domineering and selfish drives a wedge between him and the other Vees, and that's exactly what Alastor wants. So he encourages it.
The more in-depth answer is that there are a number of reasons why Vox would accept that Alastor might be impressed by this behavior:
He actually has no idea how to get Alastor's approval, so he's desperate for any kind of feedback about what he's supposed to do.
The feedback Alastor gives him is that he should be uncompromising and authoritarian.
Vox has known Alastor for decades and seen how he treats people, which includes aggressive displays and putting his own desires first.
Alastor has also been mocking Vox for being "weak" and "needing" others for decades, so Vox knows all too well that he values strength and self-reliance.
They grew up at close to the same time, so Vox could be using how they were raised as a frame of reference for what "strong" and "self-reliant" might look like to Alastor.
It's less that he thinks being more stereotypically masculine will impress Alastor, and more that the things he thinks will impress Alastor happen to be stereotypically masculine. And he thinks those things will impress him due to a combination of their shared history and Alastor himself nudging him in that direction.
Vox's initial appeals to Alastor are actually very understated, and more in line with his normal behavior. When Alastor asks what he contributes at the rally, he first tries to defend his role as the catalyst of and driving force behind the entire plan (he wants Alastor to be impressed by him on his own terms). Then, after Alastor makes it clear that being the brains of the operation doesn't impress him, Vox tries appealing to him on Alastor's terms: if he's so convinced that the other Vees are doing everything of importance, Vox will just go onstage alone so they can't outshine him.
He isn't aggressive about telling Valentino they should stay behind. In fact, he's almost apologetic at first, and his expression suggests that he even feels bad about it.
Maybe that's why he overcompensates, just a little, when he returns to Alastor and asserts that he's the one in charge.
That's when he gets his first real piece of feedback: "I suppose it's something." Which is a far cry from approval, but it tells Vox that giving orders and prioritizing his own desires is a step in the right direction. He should keep doing that, except more so. More confidence, less remorse.
Later, when Vox goes to Valentino to borrow Angel, he's unaware that Alastor is in the room. So while he teases Val about being jealous when he realizes he's mad at him, Vox still defaults to his normal behavior: first he tries to reassure him, and when that doesn't work, he switches to negotiation. But then Velvette and Alastor make their presence known, and Alastor accuses Vox of begging.
This reinforces the idea that giving orders is more desirable, but it also condemns Vox's failure to do so in a way that carries clear expectations: You don't negotiate. You don't compromise. If you want something, you take it. Anything less is unacceptable.
And that's it! That's all it takes. Vox has something to work with and he runs with it. Not everything he does is necessarily because he thinks it's what Alastor wants (the aggression, in particular, might be more a mix of increasing agitation and overcompensating for previous "weakness"), but he thinks he's figuring it out. And Alastor does nothing to suggest otherwise, because Vox is doing what he wants. He just doesn't want it for the reasons Vox thinks he does.