Can I use a single 3-conductor cable to rewire a switch in the US?
Using a single cable with three conductors (like a single run of 14/3) is the most common way to wire a switch and is certainly the right way to do it.

If using US-colored wire, 14/3 has black, red, and white.  White carries the neutral to the switch (which is required now), and the red and black are used for the switched hot.  And then of course there is the omnipresent bare ground wire.