People may not always use science deliberately, but using direct experience along with inductive and deductive reasoning is the bedrock of scientific discovery.
The bedrock of scientific discovery is reason applied to experimentation and measurement, not experience. Science requires repeatability and experience is not repeatable.
More importantly, the use of logic and reason is not restricted to science.
The scientific method is not the only way to gain knowledge, but it is also not the only method in which people perform scientific studies.
If you are not using the scientific method then you are not doing science. The clue is in the name.
[Science] does not, by any stretch of the imagination, represent the sole mechanism for understanding the world.
Yes it does. Without logical arguments there is no verifiable way to ensure you know anything. Under circumstances where too much is unknown you can use very weak arguments, but they must still be backed up by some form of logical reasoning.
Again, logic and reason are not science, they are tools that are used by science that can also be used outside of science to gain understanding of the world.