Well, to start, when you say you're afraid of the router do you mean you are afraid of damaging/wrecking the bowl or that you might injure yourself. I always do my best to avoid using any power tool in a way when I am not fully confident I can do so without injury.
I'm sure you could begin with the router if you wished. How do you plan to do it? Will you be rotating the stock somehow, using some sort of jig, or just going at it freehand? I'm not the most experienced, but I have a hard time routing in two dimensions freehand (with any precision, any way). I can't imagine doing three dimensions freehand.
Frankly, I think it's cool you can make one with an adze. Seems like that would be quiet, safe, and free from dust going everywhere. What's not to like other than, perhaps, speed?
Do you have access to a lathe?