Do you have access to a lathe, even a small, cheap one? If so, you can get the drill bit and the knob perfectly aligned/centered every time with a drill chuck inserted into the dead center side.
Another trick if you don’t have access to a lathe: tightly clamp two 3/4 inch thick boards across a drill press table one on top of the other. Drill a hole the same outer diameter as the knob a little less than the height of the knob in depth into the wood (for an API knob, measure across the wing edges for the diameter. If you don’t own calipers, it’s easiest to compare the non-fluted side of the drill bits with the knob against the end of the bit. Better to go a hair smaller in diameter than a hair bigger, as long as you can still get the knob into the hole you drill without much force. A forstner bit will work if you don’t have a big enough twist bit). Making sure not to move the boards, push the knob into the hole. It should fit snugly but not so tightly that you have trouble pulling the knob back out.
Drive a screw down into the hole between two of the wings to prevent the knob from spinning when you drill into it (you might want to wrap the screw with a few layers of duct tape to keep from marring the knob as you drill it out. Two screws between different sets of wings is even better). Now change over to the 1/4 inch bit and drill out the mounting hole.
Remove the knob (you may need to use long nose pliers with a layer of duct tape around the ends) and repeat for the rest of the knobs.