Are axes and hatchets useful for anything? It seems like knives are just better by any measure but I would like one of my characters to use a hatchet/small axe instead
Hatchets are great for clearing brush, and making camp. They’re useful in situations where you need to rapidly sever a line under tension. Switch between a blade and a hammer.
Knives give you more fine control, hatchets give you more force on point of impact. This doesn’t you can’t whittle with a hatchet if you want, people do, but it’s not going to be as easy.
So, why use a hatchet? Because it’s the right tool for the job.
I know we sometimes stress that weapons are tools. This is, usually, a little semantic, when we’re talking about a rifle as a tool, it’s because this is a piece of equipment designed to do a specific job. In this case, that, “tool” distinction is really important because, both knives and hatchets are non-combat tools first, which have combat applications.
“Why would your character use a hatchet?” Because they carry one for utility, and are in a situation where it is available and they need a weapon.
Both the knife and the hatchet occupy a strange space, they’re not improvised weapons, but they are still, primarily, utility tools. I don’t carry a knife to have a weapon, I carry one because sometimes I need a knife.
The knife is a more versatile tool. It’s an eating utensil, medical tool, has precision cutting applications. You can do a lot with a decent knife. In that regard, hatchets are more limited.
Given options, a character in the wilderness would probably carry both in addition to actual weapons. In a modern setting that might a handgun, and a rifle or shotgun. In a fantasy setting, you’re probably looking at a sword or larger axe as a sidearm, and a specialized weapon like the bow, or a primary polearm of some sort.
If a character already has a combat axe, they might not carry a hatchet. They don’t need another axe that will do most of the same things. Similarly, if they carry throwing axes, a separate hatchet will be redundant.
If you’re running with the idea of some kind ranger type character, the axe is a natural fit. It’s a good weapon, because it does double as a tool that will be vital in their environment. They’ll probably also have a knife, because it’s useful.
If you want them using a hand axe or hatchet as their primary weapon, that’s going to be a situation where, “this is what I could get,” probably not, “this is my weapon of choice.”
Also worth remembering, the knife is an excellent ambush weapon. It’s very easy to conceal. If you just need to stab someone without warning and vanish, you want a knife. If you’re getting into an actual fight, the knife will rapidly lose out to anything with a longer reach.