This derivation makes sense, but how can you know that this applies to every time measurement device?

You don't unless you assume that two postulates are correct. You may find that circular reason, but that's what your textbook is trying to teach you. The textbook is NOT proving this is true independent of the postulates.

If I'm carrying a light clock and a mechanical clock next to me, they will both stay in sync due to the first postulate.

Everybody moving relative to me MUST see my light clock slow down due to the constant speed of light (due to the second postulate) they MUST also see my mechanical clock slow down and stay in sync with a light clock due to the first postulate.

Textbook is not providing proofs of the postulate it's saying IF the postulates are true then this will happen.