Something a lot of people seem to miss about "sharpness" is that it's more a matter of edge alignment than edge refinement. What I mean by this is that you don't need to spend a lot of time and energy polishing the tool to establish a "shaving sharp" edge once you get your technique down. I find that I can start getting blades to easily push cut paper and shave (though with a bit of tug) at around 400 grit if I properly align the edge and eliminate the wire edge... In a practical sense, that means I can put a "shaving" sharp edge on an ax with a file, coarse/fine DMT diasharp, and a leather belt - and in the field, if there aren't any dings and I'm just touching up an edge after some chopping, I'd say making an axe "shaving" sharp again takes 10minutes tops. It really isn't as much effort as people make it out to be...
With that established, it's really just a question of the profile and what level of polish you TRULY need for the tool's intended tasks.
With a smaller knife, which is going to be used for a lot of PUSH cutting tasks, a polished edge will yield better results in my opinion. With my main bushcraft knife - an Enzo trapper in D2 - I'll go all the way up to a 6000grit waterstone before stropping on a .3micron chrom oxide loaded leather belt to get as polished an edge as possible. I don't use it for chopping or batoning and at 61hrc, it has insane edge retention so I can justify getting it that polished. At that level of refinement, the thing is a pleasure to work on wood with - it bites easily and slices through the wood like butter.
Now with something like a bushcraft axe - lets say my Gransfors Bruks SFA - which is used for both basic wood shaping/carving and chopping tasks, it would really be a waste of time to get a very high level of polish: the second you stop using it for woodcarving and use it for hard chopping, the impacts will ruin the polished edge quite quickly (wouldn't be worth the time you'd have invest). That said, I'm pretty content keeping it at around 600grit - still "shaving" sharp and great for shaping wood, but much more maintainable than going all the way up to 2000 grit.
Lastly, there's a dedicated chopping implement like a full-on felling or splitting ax. Really, in this sort of situation, you're relying significantly more on the blade's profile and generated force than you are the edge's level of refinement... hence, if the edge has the same profile and alignment, you wouldn't see much difference in the effectiveness of an axe sharpened only with a file and one polished up to a mirror shine. HOWEVER, anyone who's ever processed wood with a full sized ax can tell you how handy it is to be able to slice away springy branches, saplings, and other foliage that might obstruct your swing... this is something that isn't easily doable with an axe only sharpened with a file (which tends to still have prominent burs left over and not be able to "bite" into springer targets). This is why many ax-users opted to have both a file AND a stone for sharpening axes - so that the bur can be removed, and a keener, "sharper" edge established for such tasks.
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Sorry for practically writing an essay on the matter, but who knows, someone might find it to be a useful take
but growing up part of our job was to "chop cotton"! That means you take a "hoe" to the cotton field and remove the offending weeds while leaving the cotton plants. The best ones where made from old hand saws brazed onto the neck of the hoe. We carried a 6 inch flat bastard file in our back pockets and "touched'em up" everyonce in a while. I can tell you that I still have the hoe my Great Grandfather made for my Grandmother and it hit the dirt a thousand times a day (we have no rocks in our soil here BTW) during growing season and it is still a fine tool today! I only told this story to say that I sharpen everything acording to its intended use.
