I think the concept is even more beautiful than you described:
A symmetry in a physical system implies a conservation law.
As a physicist, since the beginning of your studies you learn to appreciate and seek symmetries in various systems. At first, it’s mostly on an intuitive way to help you understand or simplify a problem. But at some point you learn about Noether’s theorem and see the even deeper meaning and power of symmetries.
For example, symmetry in movement in space (meaning I can move my entire system and it stays the same) implies conservation of momentum.
And symmetry in time translation (meaning if move the entire system a through a same interval in time and it still behaves the same way) implies conservation of energy.
Indeed , more is different:
https://cse-robotics.engr.tamu.edu/dshell/cs689/papers/anderson72more_is_different.pdf