-- But let me clarify! I want to directly attack those who you refer to thinking that "if you can't explain or prove something rationally then it's not of any value". I don't think anyone is a constant calculator like that. It's important to learn because we're building computers that operate that way, but it doesn't appear that strict rationality (whatever that is) is our true operating mode.
So if you want to get the gains of rationality, the best you can do is use rational structure to train and shape your intuition. If you're skilled at the math required then you can use it to empower your intuition -- but never replace it.
I say this with strong fear of thought paralysis. Those who spend too long rationalizing every thing seem likely to trap themselves in local minima, to argue endlessly over two similar choices while missing out benefit of either.
I like to think about MCMC algorithms, actually. They're guaranteed to converge to the most liable posterior beliefs, but do so by jumping randomly. Each step forward is technically blind and hopeful, but by keeping a goal in mind and learning from every jump you improve, even considering the immense ignorance that Markov methods maintain.
Intuition can do better.