It depends on how you use your equipment. If you use it often enough that you're going to change fluids and filters based on the calendar and not on the hours, then the wear and tear on the machine isn't worth talking about. At that point, its a question of whether having the splitter mounted on the back of the tractor works well for the way you deal with firewood.
On the other hand, if you own an old, worn out tractor that's on its last leg, and you're only putting 10 to 20 hrs a year on it, then the additional wear and tear of running your splitter is probably not worth it. This is the boat that I'm in. I only gave $2K for my tractor (1983 1710) with its loader. It has over 3k hours on it its never lived indoors, the only electronics that work are the starter, temp gauge, and tach, the hydraulic pump is a little on the weak side, it has a couple of hydraulic leaks once the oil warms up, the engine leaks a little oil, and the head isn't in the best shape. I bought it as a test to see how useful a tractor would be for me. I knew I could get $2K worth of work out of it. I use it for bush hogging once a year, occasionally plowing snow off of our road (1/4 mile, dead end), dragging logs off my trailer and moving them around the yard (including bringing them up to the house to burn) and some occasional dirt/stone/mulch work. Adding a hydraulic splitter to the tractor would likely double the number of hours I run it which would cut its remaining life span in half and it's not worth the tradeoff for me.
Even if that were not the case and I owned a much newer machine, I don't really want to have to get mess with the tractor every time I want to split firewood, especially in the winter. I typically only cut and split 1/2 to 1 cord at a time (as needed, or as time permits) I'd much rather have a dedicated log splitter sitting where I do my splitting.
A friend of mine is in the opposite position even though he also uses a Satoh that he bought used in the early 80's. He has steel firewood racks that he had made decades ago. He has always "worked from home". He cuts and splits all of his firewood for the season and stores it stacked in the steel racks, and then uses his 3ph with forks to move those racks up to his house as needed. His schedule allows him to do all of his splitting for the season in a day or two, and it makes a lot more sense for him to use his tractor hydraulics to run a splitter mounted on his 3ph. He has a skid steer with a bush hog attachment, so his primary use for that tractor for the past 2 decades has been splitting and moving firewood.
There isn't a right or wrong answer to the original question. It just depends on what fits your needs the best. I will say that my friend does NOT take his splitter out into the woods. Its just another piece of equipment that he has to move back and forth. That said, I would NOT want to take MOST towable splitters out into the woods. The ground is too uneven, and the wheels are too small (not enough ground clearance). Too much risk for damaging the splitter.
