The cruise control in my 2013 Volvo keeps the speed nailed pretty much nailed on the set speed. The dash is digital and so you can see how the spedometer needle stays perfectly covered up over the green set speed marker.
They tend to use the old "fade in" control trick to get it to behave more comfortably. You don't need to instantly give all throttle control to the PID loop, but instead you can gradually ramp up how much it is allowed to affect the throttle. That way is slowly starts accelerating and it gives the PID loop some time to get settled in before it fully takes control. This trick can be used on most PID loops to smoothly switch them on without the whole thing going crazy in its first moments.
If you are to run a generator using cruise control and suddenly go from no load to turning on a 10kW flowtrough shower heater then yeah pretty sure it will not keep the the speed perfectly steady, likely dropping in RPM a lot before recovering back up, that would happen to all generators because the throttle response is not instant, so it has to rely on the inertia of the system to fill in the gap a bit, in order to get kinetic energy from inertia back you have to slow down the RPM. But as long as the engine doesn't stall and quickly recovers it is fine. The small generators indeed tend to have very simple mechanical governors and they work well enough for the job.