Any source of knowledge that predicts a real world outcome is testable by "science." For instance, if prayers can change outcomes, that power can be statistically determined.
From what I can tell of all scientific studies, if God is involved with the day to day workings of the universe -- his fingerprint is indistinquishable from that of random events that exist within the range normally ascribed to nature.
So your assertion that there are alternate routes to knowledge about the real world is questionable -- since all testable routes end up having materialistic explanations.
But I'll ask, do you have an example of an alternate route to knowledge that came to us through intuition or divine revelation and not through scientific discovery?
Well, many millions believe that the miracles attributed to Jesus Christ (and his resurrection) actually occurred. They are among the events the most historically supported. Yet because they are in the past they are untestable. Yet if they occurred, they are outside the realm of current science.
One example of "intuition" is Einstein's theory of relativity. It was theory, not proven until years later (parts of it are still being "proven").
There is a lot of creativity and intuition in persuing pure science. Theories are imagination, and then it takes imagination to test the theory.
The idea that things can be measured and that laws of nature are universal are assumptions based on "divine revealation".
The 12th century philosophers who began science assumed that the universe was logical because God was logical. Because God was logical, then man, using his logic, was doing the work of God when he explored the universe. So you could say that the entire field of science is based on a "divine revealation" that nature is not a mysterious god to be worshipped, but a creation of a logical creator. And science assumes there is a logical explanation behind nature, because the original philosophy behind science, i.e. Christianity, assumed a logical creator who created things logically.