When routing it seems important to make shallow passes and move fast enough to avoid burning. The last pass should be very shallow and fast thus removing its own burn marks. Different types of wood seem to burn differently, so doing test pieces might help get the right speed.
If routing by hand, it’s usual to go counter-clockwise around the outer edge. If the surface is curved, the router can bite into grain and cause deformed edges. You might have to route in one direction on one part of the curve and in the other direction on the other side of the curve. It’s tricky.
A spokeshave can easily and safely and quietly and smoothly round over edges. And it’s easier to determine direction of the grain.
Hope this helps.