At best this amounts to "the people don't care enough about the things I think they should, therefore there's something wrong with the people".

It could amount to this, I guess. But I don't see why you'd think this is all it could amount to at best. Do you really consider it outside the realm of possibility that people could be genuinely better off with certain social changes and yet fail to acknowledge this fact due to conditioning?

And yet you're perfectly willing to dismiss those same anecdotes as "false consciousness" if they don't support your ideas about how much impact there should be on the "lives and psyches of the population affected".

Just because I think an anecdote reflects false consciousness doesn't mean I'm dismissing it's evidentiary value. A marginalized person doesn't have to be saying "Look how I oppressed I am" in order for us to listen to them and realize they're oppressed. Judgments of oppression are judgments about the objective conditions of people's lives, not subjective facts about how they feel.

A personal example: I've volunteered to conduct surveys in rural India in the past, and this involved talking to women in Indian villages. Virtually none of these women explicitly referred to themselves as oppressed, and I doubt most of them consider themselves oppressed, because they have a host of bullshit religious and traditional beliefs that prevent that realization. But hearing about their lives, it was evident to someone who does not share those bullshit beliefs that they were in fact oppressed.

So when I said that one needs to listen to marginalized people in order to fully appreciate the impact of a lack of privilege, I wasn't just referring to marginalized people who're yelling about oppression. The only thing I'm "dismissing" (although this is probably not the right word) when I talk about false consciousness is the idea that people's subjective judgments about their oppression are a reliable guide to the objective facts.

And just to be somewhat even-handed, let me acknowledge that I think there are certain social justice communities where the unreliability runs in the opposite direction, where people are conditioned to view everything through a framework of oppression, and they overestimate the extent to which various practices are oppressive.