I have and use both, fixed and mechanical, and have for decades to kill a couple hundred deer. Cut diameter/blade height, and the vented area on the blades can make a difference. BH materials and construction makes a difference. The angle of the cutting surface can affect penetration, but not by a lot. If you put all of the factors into a calculation you get "Form Factor", and that has an effect on momentum, (under controlled conditions that don' exist when you're in a tree somewhere). If all blades, sheer angles, and cut diameters are the same, three blades could, would, might penetrate better than four blades. But not by much.
The best answer is pay attention to what you're equipment has for delivering energy to the cut, a combination of physics and experience, and buy those. I've used Muzzy 3 blade and 4 blade heads with a 1" cut Diameter on and off for ??? years. I've never seen anything that suggests there's any difference at all when you can see the critter hit the ground from your stand. And I typically can. Like the same question in the monotonous discussion of fixed or mechanical, it depends. If you're counting on cut diameter, or one more blade to make any difference, at all, you need to be more selective when taking shots.