Is there any advantage, when playing solo, to not having a companion fight by your side?
There are advantages and disadvantages. I'll outline them both below.
Advantages to going solo
Disadvantages to going it alone
Interestingly enough, The original plan from Blizzard was to have followers lose all utility after normal mode was completed, however they have changed their stance recently and want them to remain viable throughout the game. They certainly provide a wide array of helpful skills, so I like to take them with me.
A companion will automatically attack hostile mobs, so it can be useful to not have one to control when you aggro an enemy(particularly Treasure Goblins).