3/16" is the golden thickness for me, which puts it into the golok category. Much more than that and I think you're just adding weight, which can be good but then I would rather have an ax or hawk as I like chopper blades more than axes and hawks for portability so reducing the ease of carry makes me start looking at things designed differently. There is some exception to that as I've had a few knives that are 1/4" up to 0.3" with FFG or full-convex grinds that did a good job removing steel to get the weight down, but the only advantage they have over a thinner blade with some flat areas on the side is a bigger spot for a baton.
The exception to that is kuhkris, which I don't mind thick but I pretty much lump them into the hatchet category when they're thick since they can also be more annoying to carry but add a different type of utility as well.
For 3" branches and wood carving, I would be thinking 3/16" thick, 10-14" blade, if I was doing more of the chopping of limbs, maybe down to 8" if I wanted a little more control over the carving aspects and had an adequately long handle so there was a rear chopping position and a forward carving position.