there are lots of reasons to run a mower at wide open throttle and all of them are good reasons.
1. proper fuel air mix for the engine design....these are simple carburetors...not variable valve and not electronic ignition...timing and fuel air mix is best at wide open throttle for best engine performance. and also for optimal oil lubrication.
2. cooling: the flywheel has cooling vanes. at wide open throttle you are getting the best optimal cooling effect...it's slight difference, but it does matter to cooling
3. mowers are belt driven...blades and the hydro drives (PTO) require belt driven power at the same time....arguably, at higher RPM, these belts and pulley systems will produce more effective power to drive those systems. Wide open throttle allows for this. Especially so when cutting high tall tough grass and moving up inclines. Engines that are forced to work hard at lower RPM's means there is more chance of belt slip and reduced power and this results in more engine wear as the governor is engaged all the time under higher torque loads and where the PTO clutch slips. At higher rpm's under the same loads, you are going to be more efficient with less wear and more effective power.
all things being equal, air cooled engines for mowers are best powered at wide open throttle. I think most people can understand the difference between this and lets say a air cooled motorcycle engine, where obviously you are not putting a constant load on the engine on stop and roll. On mowers, you have two things going on: engaging a drive system AND turning blades. In most use cases with mowers, you are doing both at the same time, so in effect there really isn't a stop/start condition like a motorcycle, even though they are both powered by an air cooled engine. (caveat: I understand some motorcycle are water cooled...just making a apple-apple comparison)
I am sure you could mow at lower than wide open throttle, but you will also realize a poor cut, slower speeds, so any gain you might think in terms of fuel efficiency is subtracted by longer mow times. About the best thing you can do with a air cooled mower engine to get the best power and fuel efficiency is clean it up, fresh frequent oil changes, clean plugs, clean, clean cylinder fins, clean fuel, sharp blades, proper tension on blade and pto belts, and proper tire pressure, and proper valve clearance. These air cooled engines are designed to operate within a fairly narrow rpm range from idle to wide open throttle. They are designed to operate at idle rpm's to get oil and lubrication started to prevent excessive wear before high load cutting...but after a few minutes of starting, that lubrication cycle is complete and the engine can then be put into high load, wide open throttle. That's how they are designed to operate.
I don't think a lower than wot setting is going to damage hydro drives, but you are definitely not going to be getting the higher rpm and thus pressures delivered to them. So it will be slower as a result in lower than wot. And some of the higher end hydro drives have a cooling fan above...that rpm spin is directly controlled by the rpm of the pto belt drive...from the rpm condition of the engine.
All things considered, when mowing...wot for the win.