I don't go anywhere without a full compliment of tools, which includes an ax. it was my main weapon until the swamp when I began to level clubs. you need to be leveled well with two weapon types to cover enemy weaknesses and resistances. I'm in endgame and I carry Frostner, Drauger Fang, and blackmetal ax. So yes, as long as you have other options for specific situations then the ax is perfectly fine if that's how you want to play it. like you, I wanted to do classic viking with a bearded ax and round shield.

important note: I do occasionally carry the iron mace instead of frostner, especially when in the mountains. frostner splits it's damage between blunt, frost, and spirit damage whereas the mace is just one big wallop of blunt damage and better suited for certain enemies.