YeeHaw, all this one handed talk, especially the pushing chunks with the saw buried, makes me cringe. I second the idea that it is rare to have to one hand, you just have to make that concious effort, keeps you on the ropes where you should be anyways. I reserve one handers for when I know I'm safe, wood between me and the saw for example, or when it is the safer choice as mentioned.
I watched my partner one hand chunking down trying to push it over. I made sure we had a little talk, I don't even know if he was thinking about it when he did it but he agreed it wasn't safe. I carry a crowbar for the big stuff, flip the hook up through a ring on your harness and forget it's even there, just in case there's a little more hinge than you thought.
As far as spiking a leaner, they're not easy to stay on top of whether a limb or a whole tree, I always go out of my way to stay there. Even taking the time to dbl rope using the end of my climbing line in a pinch.
It's tough, we're all faced with doing the safest job as quick as possible, an obvious conundrum (vocab toilet paper).