The problem with relying on rationality and logic entirely is that it is, ultimately, binary: true or false.
This fails to capture mathematical reality. I’m not just talking about Gödelian incompleteness, although that does factor in. But real numbers can only be approximated by bits. There is an infinity of real numbers in between any two real numbers you can think of, but a computer has to round to a terminating binary expansion (also known as a terminating bicimal) at some finite precision.
So all computer programs are necessarily approximations of mathematical reality.
Binary represenation also fails to capture physical reality, since quantum mechanics is probabilistic. While quantization means that there are a finite number of states, Heisenberg’s Uncertainty Principle precludes probabilities of 0 or 1.
So I am generally inclined not to accept rationality and logic as the end-all, be-all of epistemological methods.