That sounds strange to me, and radically different to the way I was taught!
I was always taught the only thing you do before the first flight of the day, that you don't need to do subsequently, is check for water in the fuel tanks that may have accumulated over night.
My instructors have always told me to do a runup before every takeoff, even if you do a full stop landing and taxi straight back to the hold to takeoff again (e.g. when you're doing night rating circuits). As you say, the point of a runup check is to ensure the engine is operating correctly before the most critical phase of flight.
I have seen pilots at my club start up, taxi out and takeoff without a runup but certainly wouldn't be comfortable doing that myself (or even flying with a pilot who thought that was acceptable).