> Are they actually trusted to give independently good advice, or are they trusted to launder and expand on advice that leadership wants to hear but can't say themselves so that it can be plausibly deniable when it goes South?

It's both. There's some expectation that they aren't actively self-dealing... Besides the obvious expectation that they'd love to sell you further services. That's normal and expected, whereas them shilling for their other clients is not.