A bowl bit of the proper size will, usually in two strokes, will provide the rounded bottom and the oval shape. A Forstner bit is a drill bit which leaves a groove, easily cleaned up, and a flat rather than bowl shaped bottom resting surface for the pipe. The router bit will leave a nice oval dish shape.
The rest hole can be shaped (engineered?) in various ways. I prefer the router as it is not a tedious job, leaves a fairly clean depression which requires little cleanup. The edges are sharp. Various hand tools such as gouges and chisels can also be used for the "scooped" shape. I find such work to be tedious. Others enjoy crafting the rest entirely by hand.
Quick and dirty with drill bits or a little more time with a "finished" look with the router. Choices! Always choices!