A generic valve may not have the ability to properly refill the tank and bowl without either shorting one or the other, or wasting water down the drain.
Almost ALL fill valves are "generic". Most are either made by Fluidmaster, such as Kohler, or Coast for everyone else. I use the Fluidmaster exclusively, except for "lowboy one piece" units. I use Korky flappers for ALL replacements.
Some toilets with low tanks will take a Korky 528.
I use both the Fluidmaster and the Korky, depending on which brand I'm working on.
They both work pretty well in a generic situation.