Like, for example, if I built a justification for my believing in the existence of souls on top of the belief that the human mind is uncomputable, then if I turned out to be incorrect about the human mind, it might lead me to believe that souls don't exist (though that wouldn't really be a logical result, but I am not always rational). So, in order to have better foundations for my personal beliefs, I prefer not to rely on the assumption of uncomputability of the mind (nor on its negation).
Basically, I want to avoid being too optimistic and believing too much because of it being convenient for my other beliefs.
It does seem likely that ideal minds would be (for reasons like those you describe).
I just think I should require a rather strong argument to conclude that they are, instead of remaining non-commital.
I don't see what will come "crashing down." Science is about being able to quantify and empirically test our hypotheses. Nothing about uncomputable intelligence seems to undermine this idea, but perhaps I am missing something. Instead, it broadens the range of hypotheses we can use to explain the world, which seems to be a good thing.