To a degree. Initially as you lose weight, riding at X mph will require less power (moving a lighter object!), so you will sweat less.
But once you reach a reasonable weight and are just gaining muscle and cardio power, added fitness level just allows you to emit more power for longer. But sweat levels are pretty much just related to power (how much heat you generate) - and of course cooling.
At Grandparent's speed (22.5 mph) on flat ground, he's going to produce massive power to overcome air resistance. The effective fan will help cool him down/dissipate sweat, but especially in body areas that don't get such a good air blast, he will end up soaked.