@layman,

Of course theoretical physics takes existing theory that has been validated by experiment to build new models. Experimental physics then takes the new models to find a way to test them.

Both of these parts of physics are mathematical. Find any original scientific paper (rather than a description of the science written for lay people) either theoretical or experimental and you will find it is based in mathematics.

You are wrong about special relativity.

Special relativity was introduced by Einstein in a paper entitled "Zur Elektrodynamik bewegter Körper" ("On the electrodynamics of moving bodies"). You can read an English translation of the original here

http://www.phys.lsu.edu/mog/100/elecmovbodeng.pdf
From the very beginning Special Relativity (like any other Physical theory in the past 500 years) was based on a mathematical argument. It was proven to be a valid experiment when the mathematics of the theory made correct predictions when tested by experiment.

This wasn't something that was just made up by idle musing. Einstein studied extremely hard to master mathematics and to understand the work of the Physicists before him. He used rigorous mathematics to develop his theories which held up to the scrutiny of his peers.

Your idea that Einstein was just some random guy who came up with an idea is wrong. He was well trained, educated, studious and exact in his mathematics.