There are several router or shaper bits (A shaper is a declining use tool in home shops). Youd need to use the router table with the shovel handle secure and then you would "feed" the end stock into the bit until it mde the cuts.
Id suggest looking at a larger "Round Over" bit.
Also, Ive done blunt ends on walking sticks or home made chisel handles by using a bench grinder and a homemade "jig" that would hold the stock securely as you feed it toward the grinder stone.
You could imagine such a jig as a kind of "Cradle" in which the stock could rest and the cradle could be secured to the bench grinder table using hold downs or small clamps.
The only thing against using the grinder is that the wood tends to burn a bit, but that can be sanded off easily.
In order for you to make this look smooth, you need to keep your work stock securely held against any grinding stone or even using a table saw blade, (The use of router bit and table also requires the stock to be firmly held against the blade and the infeed be done in small "bites")
There is actually a tool that puts blunt ends on to canes or pointed ends for making rustic stool legs. You would feed the end stock into such a cutter and crank it by hand like a pencil sharpner. (It kind of looks like a thread cutter) I know a guy who bought one (I think from Rockler ) and he makes these rustic stools where he leaves some of the bark on the stool legs and makes a carved dowel like legs and drills holes to receive them . He sells the stools at craft shows .
I couldn't see buying such a tool for one one or two rustic stools and then never using it again.
If youre going to make these things as a routine maybe you should look for one.