It might ‘be a pain’, but the quickest, easiest way of getting a usable edge on a basic working axe is a file, or a couple of pucks.
You can make your own, if you’ve got a spare round fence post, just cut some slices off, around an inch or so thick, making sure the cut faces are dead flat, sand them smooth and varnish them, then get several grades of wet and dry, and a removable adhesive, spray the flat face and place on the back of the wet and dry with a weight on it, when dry, cut around the wood with a utility knife. You could make a whole bunch, going from around a 100 grit, then working up 200, 400, 600, 800, 1000, 2000, but for general work no higher than about 800.
But really, a couple of good 2nd cut engineers files, a coarse, medium and fine, that should work.
There’s always a Lidl bench belt sander, but that’s a good way of destroying an otherwise good tool if you’re not very careful.
We’re not talking fancy spoon carving tools here, just a basic hatchet.
There are diamond ‘stones’ you can buy, but good one’s ain’t cheap, wood glue and sheets of wet and dry are.