my $0.02
To set a wedge, a hatchet is fine.
To "drive" a wedge I want a full sized axe or 8#+ sledge hammer.
I guess it depends on what size and type of tree. I consider directly behind the tree, barber chair territory, and I do not want to be there tipping a tree with a tiny implement using two hands. Though I many times wrap with chain or strap anyway. I want to at least be arm's length away.
Kneeling on the ground, for a 8" DBH tree with <5" wedges, I would go with a hatchet. 15" tree 60+ feet tall, with hard head wedges, full sized axe or sledgehammer.
I conceal carry a pistol almost 3 pounds heavy. I can feel it after a day of outside work, so, I would put that into the mix if I wanted to carry 48oz on my belt for a full day of work. You might find a full sized axe stuck into a carrier on your back better for a deep walk into the woods and much handier.
I tend to lean whatever I use right next to the tree I am cutting so I am ready and not searching trying to pull something out under pressure. I tend to do almost all urban trees, so, I take a folding step ladder, lean my sledge against it, and the various wedges on each step, so, I can quickly choose. Start small, end big.
Many big oaks require 6-8 wedges, I am not carrying all that on a belt. I put them into an orange plastic tool box along with a drilling hammer. I carry that and open near each tree when doing a lot clearing. Going to need more then 1-2 wedges, pre-stage them, easy to grab by the non-wedge end, sticking out or on top of the box.
SAFETY
Exiting the danger zone while dropping trees, I have fallen before. Good reason not to conceal carry behind your spine. Fallen before on my side carried gun, not sure I would want that to happen with an unsheathed razor sharp hatchet.
If not carrying a gun or two, a hatchet would be something I would consider a must carry in the woods, easily accessible, along with a full tang knife. I rather carry the wedges and hammer/hatchet in a box and keep my water bottle on me or a machete.
Knocking over <6" birch in Sweden is different then knocking over 32"+ sweetgum in Georgia.
Taking a hatchet to tip over a 32" Georgia sweetgum with wedges would be like taking a .22LR or 9mm to hunt bear. Yes, you can do it, but, it might not be the best tool for the job, even though it might be easier to carry around the waist for the short term.
I cut down a bunch of trees at Iron Gait Percherons, small video of them running in the background
A short video explaining how I cut and drop trees with a short video taken at a horse farm.
rumble.com