So here's my concern. To properly use a cutting tap, one must start the thing very accurately. One of the reasons you use a tap to chase threads is when someone has run a pedal in at an angle (that is, they've cross threaded the hole).
Chasing threads is a time-honored technique in machine work. But the taps do remove metal. So you end up with a weaker, looser threaded hole. Furthermore, if the person doing the threading is a novice (or even an momentarily inattentive experienced guy) and the tap gets started wrong, you've made the problem worse. Running the tap in from the inboard side is a good idea: the inboard threads probably aren't cross-threaded and can guide a good start.
But here's a question: another type of tap is called a form tap. Instead of having cutting edges, form taps are smooth, and have threads of gradually increasing size. In essence, they form new threads as they are screwed into the hole. There is yet another tap form, called a "chaser", which cuts a bit less metal than a true cutting tap.
Does anyone make chasers or form taps for 9/16 x 20 CEI LH and RH threads?