Frank,
Sounds like we have a different style of camping and different climates. When I am camping splitting wood is not a maybe, its a definite for me.
Here, you are not going to find much dry wood on the forest floor, except maybe in the middle of summer, even then much of it will be wet.
Mostly I use hard woods like maple, oak and elm. There is of course plenty of pine, but it is the worst wood for my uses, it hardly throws any heat, is smoky and burns way to fast. Time spent needlessly tendering a fire is time that could be spent doing more important things, like napping and eating.
I usually select a camp site that I will use the whole season, coming back to it on weekends and when I get a chance during the week. Some sites I have used for several years, or come back to a few years later. I sometimes come back and find that other people have used it, which does not bother me unless they leave a bunch of cans, paper, and other litter for me to clean up.
I usually build a lean to shelter using birch poles for sleeping under, and I pitch a tarp next to it to use for a food prep and craft area. When it is raining lightly, I use this as my kitchen if I have my MSR Whsiperlite gas stove. I always keep a little supply of kindling under my tarp so it is dry when I need it, and ready to go. When my family is with me, they we use tents and use the lean-to as a play and craft area.
I find a dead standing tree, usually about 4-6" in diameter, near my camp site. I take it down with a saw, then buck it up there into manageable lengths. The thicker the wood, the shorter the lengths otherwise it gets too heavy to easily carry and move.
I always use a saw for this. Well seasoned hardwoods like Elm can be a bear to chop through, you just are not going to get the kind of penetration you do with green wood and soft woods. As well, vibration and shaock take a toll on your shoulder and arm, and the hard wood can really stress a thin edged axe or knife if you are not very careful.
A good saw eats right through the wood. With a good bucksaw, I can buzz through a 4-6" round in under a minute. While I might be able to do the same with an axe, it would be a minute of seriously hard work which I try to avoid. Using the saw it is a leisurely pace.
For limbing, I sometimes use a big branch, or the spine of my knife for the small stuff, if it is small and hard it will shatter. This stuff can be hard on an edge, and while I have no doubt my knives can take, it is the same amount of work to break them off and it saves sharpening time. Bigger limbs, I use a big knife and lop them off, though if it is thick and hard enough of a limb, I just use the saw to prune it off. Basically, i do whatever I think will be easiest, or in the alternative the most fun.
I pack the wood back to my camp site and if there is still a good measure of bark on the tree, I will strip enough off (which sometimes is not easy with dead wood and other times just falls right off). I set the bark aside, it won't burn as it is usually very wet, I use it to put on top of my split wood pile, shingled to help keep the rain off.
I use the first round (nearest the ground and usually the thickest and heaviest) as a block, and use my big knife and spilt the logs into useable thickness. On a 6" diameter log, I will usally split the wood several times to make wood small enough to get fires going and for quick lunch fires, etc.
Splitting the wood is not that hard hard work, I use a rather heavy mallet like baton and go slow. For me, this is much easier than walking all over the place looking for wood, carrying it back to camp, etc. Also, when using small woods (finger thickness stuff you find), it seems that I am always on the search for wood, and this takes away from my napping, reading, playing with my family and dog, eating time, etc.
For canoe portages, usually the trails are well cleared, as I am not in any super rural spots. In fact, in most areas I am in, I can still get cell phone reception, which may seem silly to some, but when you are camping with your family, it becomes very important if there is a medical emergency.
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