Frequency and voltage regulation is totally separate issues. Frequency is RPM related and voltage regulation is done with field excitation.
RPM would need some sort of PID control.
Some sort of stability criteria is used before there is an auto transfer. You don't want to have the generator connected when your starting it, etc. Oil level usually a first order criteria for starting. Ideally, you don't want the generator trying to regulate or connected to a load until it starts and is at frequency and voltage.
Although not directly related, you have generator control panels:
https://www.dieselserviceandsupply.com/Generator_Control_Panel.aspx
The kind of device your talking about is more like an ECM,You have been talking about a frequency regulator. Is the voltage regulator bad too? Is this a single or 3 phase generator? Are you planning some sort of autostart? Some sort of output OK? Is there a battery or is it self-excited?
Simplistically, a lawn mower engine, usually the throttle is full when starting and the mechanical governor prevents overspeeding. The generator would want to control RPM.
Car engines have a turn off the fuel mechanism when the throttle is to the floor and your cranking.
Then there is choke control like automobiles. You don't have vacuum and later engines had a choke unloader, of sorts. Early engines didn't, but you could "throttle" the choke. Then you had the fuel bulb that injected fuel into the carb.
so, there is probably an off, start and run throttle positions and there could be two off states. Fuel off an cranking and fuel off (flooded)
The choke is very much temperature dependent with an unload function. unloading the choke was to position it slightly open once the engine is running. That was like 1/8" in a lot of cars. Full throttle open the choke fully. Pressing the accelerator once set the choke position depending on temperature.
I've set up plenty of carbs. Later I used the propane enrichment method exclusively to set the mixture.
Your initial post was control generator RPM and not much else. Later, you sort of assumed thatRPM keeps the voltage in check. The AVR keeps the voltage in check.
An automobile has a few modes for RPM: off, start, open loop, idle, power, economy and cruise control.
It also has a charging system which is akin to the AVR of your generator.
Aside:
In some simple generators without a battery, residual magnetism is used to create voltage for the regulator usually via a separate winding until the unit gets up to speed.
The idle is a separate system because of temperature and the AC being on.