If you're going to load a cart then you'll need something to pull the cart, like a tractor. If you have a tractor you can use it to load the cart.
We made and used for many years a timber scoot. It was what the old timers used instead of a cart. A cart means you have to move the log higher up off the ground in order to get it on top of the cart.
A scoot is basically a cart without the wheels and it is a sled without sides.
Here is one I made several years ago:

This sled was not made for hauling logs.
To haul logs you would add some brackets to the outsides of the runners for some stakes to drop into.
Then after felling your tree, and cutting it to length you would position the log scoot next to the log or drag the log to the logging trail where the scoot was and align them side by side. If you have a tractor this could be done. Then using a chain or strong cable load the log onto the scoot by using what is called a rolling hitch, up some ramp blocks/timbers. You could make your stakes strong enough to be the ramps.
Once the log is loaded we would put the stakes in the pocket brackets (made of metal) and hook up the scoot to the tractor and drag it out to the roadside pile.
The scoot is constructed with pulling points on both ends so when you unload it at the roadside pile you just hook up to the back end and drag it back into the woods.
In the above picture the scoot runners are softwood and the cross blocks are hardwood. The runners have steel shoes attached to them to prevent the softwood runners from wearing out. But you have to be careful of the steel runners hitting a rock in the woods and creating a spark that can ignite the dry leaves in the wood.
We used a back hoe with a front end loader on it to lift the logs off the ground carry them to the scoot and unload the scoot at the roadside pile:

This is a shot of my backhoe carrying a long log (here in my sawmill yard). We would use this method to lift logs, thereby keeping the clean and not dragging them in the dirt, and carrying them to the scoot where we could load several logs and then drag the scoot to the pile.
Jim Rogers