Don't get me wrong..., I've used a bow saw, a chain saw and an axe most of my life..., mostly for firewood. I've got a nice, pretty, *expensive* GransFors Felling Axe..., razor sharp, nice and big and heavy...., but we weren't talking about a specific survival situation, the person who began this thread specified "primitive living".
Yes, I know some Native Americans built large lodges, and sweat huts, using good sized logs. I've watched Cherokees burning out a dugout canoe, then chipping the charred wood out with stone-tipped tools. The Cherokees used woven cane strips, covered with clay mixed with straw.., for a kind of a stucco hut.., for their housing.
Plus, I live in the SouthEast U.S., so I don't usually have to worry about falling through the ice.
There're usually pine trees around, and you can often find a ball of pitch or, preferably, light'ard wood (that you can kick out of the rotted outer tree), that'll start a fire in a hurry. If not, you can start with the thinnest pieces of wood available or, at worst, whittle a few fuzz sticks from some standing wood.
This is obviously a very arbitrary scenario.., but, personally, I'd still prefer a knife. A knife is something you can have with you all the time (are you gonna carry an axe while gathering wild foods, or hunting?)
As has been noted..., I sure wouldn't want to skin a squirrel with an axe, not to mention whittle a figure-4, or a wooden spoon, or whatever.
But I'd certainly prefer to have both... along with a whole bunch of other stuff