Lots of good points.the main reason behind using an axe is it's versatility, but in the end it is just a tool yes you can carry a chain saw, but then you need fuel/batteries for it. you can carry a bow saw but while a saw CAN split wood, the axe will do it better. A single axe can do the work of a saw, a splitting wedge and hammer, (if you are really good with it) a knife, in a pinch it can be used as a digging tool (though MOST would consider that to be abuse), it can be used for driving stakes, etc. In the end it largely comes down to a matter of personal preference. When considered to be an essential camp tool, the truth of the matter is said tool is not really "bring an axe" it is "bring something capable of splitting firewood, felling a tree, driving tent stakes, etc". You CAN bring several different tools to do all those jobs (and more), or you can bring an axe (and yes a hatchet is just a small axe, so you are carrying a form of axe in your camping gear... it can do anything an axe can do, but may require more or less effort to do so depending on the task.) personally on foot I carry a hatchet while in a vehicle (canoe or truck) I carry a mid size camp axe (as well as other tools in both cases)
Most of all I like the reliability of an axe. Moving parts and fluids introduce all kinds of failure modes, many of which are difficult to overcome without substantially more resources than are needed to maintain an axe.