In the long run you want to do three things:
One: immediately arrest the initial yaw ( heading change) toward the bad engine. In the short run you can view the rudder as being the most important control for this purpose, but in the long run other things play important roles (see below).
Two: eliminate all sideslip so a yaw string (if it were present) would be centered. This is accomplished by putting the rudder in the position that almost centers the ball, but not quite. Typically it should be left about half a width toward the good engine. The reason for this is given in a related answer (link to be added.) This ball position should held while turning as well as while flying in a straight line.
Three: adjust the bank angle to eliminate any heading change that may have developed after accomplishing step 2 (unless you are trying to turn.) This may be accomplished with the ailerons. Given the condition established in (two), this will require a slight bank toward the good engine. After all, the slip-skid ball acts like a bubble level (in reverse- so a pendulum really) whenever heading is exactly constant, so it is clear from the ball position what bank angle must be required to hold the heading constant. The reason for this is given in the same related link noted above.
Functionally, the same result arises- at least when the goal is to simply fly in a straight line-- if the pilot uses rudder to hold heading and aileron to set the ball position, but this is really kind of a backwards way of doing things. Just as it is possible to fly blind using a turn rate indicator using the ailerons primarily to control the needle, and the rudder primarily to control the ball, or vice versa. Most would agree that the former is the better way, but both can accomplish the same result, due to feedback loops.
Anyway, anything that prevents the plane from rolling toward the dead engine is a good thing-- once a turn starts, it tends to continue- so adding an aileron input early in the game doesn't necessarily hurt anything and will probably help a least a little bit, even with no rudder input, as long as the aileron design is a style that doesn't create appreciable "adverse yaw".
I guess this answer doesn't really address why this particular pilot did what he did-- why he omitted any rudder input-- lack of recent training, poor training, poor understanding, too much time spent flying maneuvers that only require yoke inputs and not rudder inputs, or ???