I use stone axes forever.
Unless I have mass amounts of upgrading to do, I prefer it over the nail gun. I tend to upgrade things before I want to on accident with that.
Since I don't care to constantly switch out gear when going between building/looting/horde moments, I just keep it on hand and rarely bother using a fireaxe. Even if I have a good number of repair kits, I usually play solo and don't have someone to help replenish those... and you tend to burn through them quickly looting and going on missions with the fireaxe. If there were less things that could be one-shotted with a good stone axe, I might feel differently.
I also tend to put the burning shaft on it because in most situations, I don't like that mod. However, sometimes there are perfect moments when using one would be a great idea, so I justify using up a slot for a stone axe by considering it a special melee attack.
I put the stone axe away when farming trees. After that, it goes back in inventory.