Ok please do not attempt using a Wagner safety planer to cut binding channels!!!!!!!!!!
What it all boils down to is finding a way to cut a clean channel that bottom edge is perpendicular to the rim and the inside edge is parallel to the rim while maintaining a precise depth form the top or back
There are several ways to maintain width of the cut. This can be done with a bit and a series of specially sized bearings that each bearing provides a cut width equivalent within a couple thousands to the binding stack width. This system works best with a fixture that mechanically or structurally maintains the center axis of the router parallel to the rim.
No these are not available anyhere the bearing are specially sized diameters to match the cut depth needed for the lutherie industry. The one you will find in a hardware store the bearings are common sized diameters for general wood working.
Another method is to use a router with an adjustable bearing guide. This does the same thing as above by changing the distance from the adjustable bearing edge to cutting edge of the bit. An off take on this is an adjustable 2 bearing guide. The idea behind this is the two bearings ride on the rim thereby maintaining the center axis of the router parallel to the rim.
The deal with the donut on the base of the router is that the top is not a flat surface, so if the full router base rode on the top the bit would be pitched in causing a cut that is not perpendicular or parallel to the rim because the routers center axis would be tilted out at the top and tilted in at the bit. Then add to this that the plane formed by the top is ever changing the amount of tilt would be ever changing. Typically by about 2-5 degrees. With only a small area of the donut riding on the top the foot print of the top has little to no effect on the pitch of the bit.
So the really big thing that is terribly important to know is that no matter what type of jig or fixture you use or even it you are doing this hand held the one thing you must do to have good channels is to maintain the router axis parallel to the rim at all times.
Hope this helps