I guess it's a matter of interpretation, as I would think "caring about and helping the little people" is more about directly providing aid to them rather than getting revenge on their oppressors. Yes, your examples do show that Sera is connected to the little people, as they are her sources of information and lead to her and the Inquisition getting resources as a result. I don't recall any of her missions being about providing relief to the little people rather than just revenge. There was one war table mission where we waylay a caravan and can distribute its goods, I think, so maybe she gives approval for that?
But she seems to be all about getting back at nobles to avenge the little people, but that has no tangible benefit to the little people other than their gratitude. Why doesn't she have a mission where the focus is to steal supplies meant for antagonists and deliver them to a village? She seems just as much an opportunist as Vivienne, using the Inquisition to further her own agenda, she just frames it as the opposite side of the power spectrum as Viv.
True, the game is consistent with portraying Vivienne. My issue is that in order to have her be so smart and manipulative, everyone around her is dumbed down to let her get away with it. Why does the Inquisitor at no point ask her to actually deliver on her promises and provide resources? Why are we not allowed to kick her out once she's been recruited? And honestly, she should have had a much larger presence in the Winter Palace. It was such a waste that she was treated exactly the same as the other companions on her home turf. As a character, she really felt like Bio built up this persona, dropped her in the game, and then forgot to actually connect her to the world.
I think Sera could have had a great role in the Freeman subplot, as that is all about disenfranchised little people caught in the Civil War and them taking control over their fates. I'm surprised she didn't have more to say about it or more involvement.
Never mind kicking Vivienne out -- I expected her to just leave. She disapproves just about everything, and is a miser with approval. I'd love to see her gone, too, so I do think it's odd that the only way to avoid her is to never invite her in the first place. And since that was something several of my inquisitors would have wanted to do, but couldn't, you make a good point that her "skill" at not getting caught in The Game is somewhat contrived. If her power hunger is so obvious to players, then the inability to call her out on it and/or stop her is bad.
Sure, Sera could've been used more. I don't think anyone expected her motivations to be under this level of scrutiny, I'd guess. For the most part, the game seems content with backing up verbal claims from characters with their approval system. Iron Bull loves fighting, and he approves essentially whenever you fight. Varric hates red lyrium and approves whenever you destroy it (or defeat red templars). And Sera claims to care about "the little people" and grants approval whenever you do so. She never once approved the killing of anyone that didn't directly wrong her or the inquisition, and usually prefers the nonlethal approach -- pranks -- to get back at others. I think she recognizes that the problems she cares about are either inevitable due to the way society works, or big enough that even the Red Jennies cannot deal with them alone.
But on a different note, or perhaps a more important one, Sera grows throughout the game. She is childish and immature when you meet her, and that irreverence will never completely go away. But the right interactions with her get her to change her views on things, to accept that there's substance to the stories and religions she has brushed aside her whole life, to recognize that people can change. The whole focus of the cookie scene is Sera recognizing that she was tricked into disliking something in life, and wanting to redeem it.
The truth is, when you confront Sera directly about why she joined the Inquisition, she openly admits it's for personal reasons as much as anything. She is exceedingly cynical/skeptical about a great deal, but knows that seeing the world and encountering new ideas is ultimately for the best, even if it's frightening. She loves when you help her out with the pranks, but she'll stand by you even if you don't play that game (as long as you don't flip out at her about it). And for what it's worth, those pranks aren't always just about harm. The justification she gives for the Skyhold jokes is to raise the soldiers' morale.
You're not wrong about Sera's initial behavior -- she is immature when you meet her. But she matures a great deal with a strong, guiding hand. Work in progress? Sure. But then she's not the only one. Leliana is similar: she is quick to lash out and respond harshly to things. She can learn to be more careful, less violent, but it requires patience and sometimes strictness. I wonder sometimes how few people actually soften Leliana, because it's very easy not to.
Anyhow, I guess that's just how I've come to see her. She eventually became my favorite companion in the game (not to mention favorite LI), so I tend to chafe a bit when I perceive dislike of her to be rooted in not having given her a chance. They definitely made her a harder character to see depth in, but that's because she's like a real person who is heavily guarded and puts up a front to keep people at arms length. Not everyone has the time or patience to bother with that, and first impressions can be quite strong. But with all the time I've spent with her in the game, I really do think that the good intentions and heart are there to be found, and that it's not simply childish pettiness driving her.