That's a way of thinking about science, but I would argue there are more useful ways. Question-Hypothesis-Experiment is a more useful way of thinking about it if you want to generate new knowledge (do science).

> you had to commit to set of certain shared beliefs to derive your logical scientific conclusions

This is more of a description of math than science.

But it is description of both (after all, all our science is based on math). But it comes even in more basic forms: with any scientific endeavour you take a lot of things for granted and do not validate them. If you try to validate them, eventually you get stuck (as an exercise, try to devise experiment determining curvature of the Earth from the first principles). On top of that, fundamentally, theories cannot be proven, but only disproven.

All this requires tremendous amount of goodwill and consensus to make any progress. You laugh at flatearthers, but they are just a fringe example of how entire system works. At the edge of physics essentially everyone is in the similar position. Many, if not most, substantial physical advances were done by heretics working against the status quo. Of course, I think flatearthers are wrong, but I would rather have flatearthers than all alternatives from mainline eliminated.

> all our science is based on math

I disagree with this. A great deal of our science is based on math. Physics is certainly extremely mathematical. Building theories in physics is very axiom-based.

But mechanics is not the only kind of science. Much of science is highly empirical. We don't reason out our experiments from first principles. We have some assumptions and we try to address them.

An example of what I mean is the study of microbiology. Of course there are examples of fundamental ideas, like selection pressure, but "axiom" is a term of art that I just wouldn't apply here.

Two-word description of the scientific method:

Guess. Test.