At the heart of OP’s question is when does a riding mower run out of power?
Mowing? People’s grass usually doesn’t get that high. If I’m mowing tall grass and weeds, I don’t usually have to keep speed down because of (lack of) power, IMHO, it’s because stalks need time to get chopped up and discharged (i.e horsepower doesn’t change max. blade speed. However, if you hear engine bog down, you can say horsepower is a factor.
Going up a hill? IMHO, I doubt many people mow that fast or have that steep of a hill that they run out of power.
Towing a cart? Yes, going up a hill with a loaded cart, horsepower will get up quicker, same with accelerating. On flat ground a smaller horsepower will accelerate to the same top speed as the big Hp, just at a slower rate.
I can’t think of anytime mowing with a “18hp” mower that I’ve run out of power. There’s been times snow blowing when rpms dip because of load, but again, like mowing, more often one has to slow because the volume of snow overwhelms the rate (at operating rpms) the blower (or mower) can process; not because you run out of power.
...the belts and pulleys are such that mowers and snowblowers can only process so much material at their operating rpms. When you overload, people will slow down because of this, not because of lack of power. Sort of self correcting.
And when you slow down, you don’t need additional power.
Quick physics lesson: say you want to climb a 100’ hill on a ten speed bike.
It takes the same amount of Energy (or work), no matter if you do it in 1 sec, 10 sec, 1 min, 1 hour, etc... , or use a 1 hp, 2 hp, or 10 hp motor.
Energy (or work) is a total. Takes the same to get up 100’ no matter what.
Horsepower is a rate (of energy) that you can do work.
(Say) at 1 horsepower it takes 2 minutes (120 sec) to climb; at 2 horsepower it takes 1 minute, at 10HP it takes 12 sec.
That’s horsepower.
Now looking closer at the 2Hp in 2 minutes climb. If motor can do it in high gear at 100 rpms, that’s 10x the torque of a 2Hp motor that runs at 1000 rpms for the same 2 min climb. That’s torque and gearing.
Realistically, it gets more complicated because motors put out different torque and horsepower at different rpms; gear ratios have limitations, etc.. Also a machine’s weight (and traction) usually mirror the HP (so a 20hp usually can’t pull the same load as a 200hp tractor if it goes 1/10th speed...but sometimes it can).