I think that ''evolved faulty thinking processes'' is the wrong way to look at it and I will argue that some biases are the consequence of structural properties of the brain, which 'cannot' be affected by evolution.

The structure can be affected by evolution, it's just too hard (takes too many coordinated mutations) to get to a structure that actually works better. I think you recognize this by your use of scare quotes, but you would be better off stating it explicitly. This is the flip side of the arguments I think you're alluding to, that the faulty thinking was actually beneficial in the EEA.

There must be an evolutionary explanation for the properties of the brain, but that doesn't mean we need to actually figure out that evolutionary explanation to understand the current behavior. Just like there must be an explanation in terms of physics, but trying to analyze every particle will clearly get us nowhere.

In fact, if you can find an explanation of a phenomenon in terms of current brain structure, I think that screens off evolutionary explanations as mere history (as long as you've really verified that the structure exists and explains the phenomenon).

I do think we're getting sidetracked by your halo effect example, though -- it might be useful to give three or four examples to avoid this (although if each one has a different explanation, that might substantially increase the effort of presenting your idea).