Every liberal does it too, from center right radlibs to far-right "conservatives": the most extreme right fringe liberals hate the mainstream liberals for not being bigoted enough, the mainstream libs hate the radlibs for not being cruel enough, and the radlibs hate the left for not being chauvinist enough.
Denouncing chauvinism in particular is like a liberal moral event horizon, a cardinal sin against their self-interested belief in the righteousness of the imperial hegemon that keeps the treats flowing at gunpoint.
I had mormon missionaries try this tact with me, that at some point 'deep down' I knew that their truth claims were correct. I don't consider this a workable model, as it basically gives up the game, and forces us to work backward. 'Knowing deep down' or 'at an emotional level' is a common colloquialism but it seems to reifying a sort of compartmentalized cognition where there are multiple cognizers within and individual knowing different things and not communicating, which I don't buy.
I'm not even convinced cognitive dissonance is an actual thing, as it presupposes humans have a need for logically coherent world views, and given that 99% of humans are not even acquainted with the rules of formal logic nor do they use them in arriving at their individual beliefs, I don't see any reason we should experience discomfort when those rules are broken by our beliefs. The psychological studies that I've read that seem to show cognitive dissonance exists could just as plausibly show that people get uncomfortable when their views are publicly interrogated under some formalized system, which is already pretty obvious.
We know they know we're right, though, because they'll tell us when our ideas are separated from the material changes that are required to implement them. Ask a lib what they think of:
You're probably going to get supermajorities on roughly the same page as us, at least when they're speaking in the abstract. They do value many of the things we do.
The problem, of course, is when someone whispers "property values" or "the price of gas" in their ear. Suddenly all those good intentions vanish because it could conceivably be a threat to their own material conditions. We know they know that's wrong, too, because libs lionize people who put what's right over what's profitable.
I don't disagree that they agree with the moral righteousness of any policies (or oppositions) in a vacuum, they just deny that vacuum exists, which they are of course right to do. Take universal healthcare or public housing or anything else, their position is typified by saying "it would be great if we could have this, but we can't, because there would be worse moral tradeoffs if we did". The quintessential examples for this are universal healthcare stifling innovation and immiserating hundreds of thousands of workers.
Now are those objections true? No, but they believe them, and that's how they can see their position as being the more moral of the two.
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The cognitive dissonance thing is something I've been mulling on for a while as well, kinda neat to see a take on it that helps me sort out some thoughts.
People abuse it for their convenience and it oresents epistemological problems, but the dynamic you seem to be describing g is absolutely a real thing. Minds are not indivisible.
That's putting it lightly. Brains are not indivsible, and you can absolutely overlay a divisible conceptual schema (id, ego, superego, whatever) onto the mind, but as a functional unit for interaction and access, minds are indivisible. I can't walk into another part of my mind like I can walk into a different room. If I could, who is doing the walking? This model seems to posit a sort of Inside Out scenario where we're actually a constellation of different cognizers, which I don't buy.