Oh GURPS....
From my rusty memory of WAY BACK when it was released...Generic Universal Role Playing System was the Steve Jackson take on the generic RPG. Generic rules, with background / world splat books to fill in the blanks.
You want to do sci fi/ fantasy / Horror....etc...same rules, different world books.
It was a point based buy system for attributes. I remember my beef with the system was the attributes seemed overly simple (only 4 IIRC).
It will be crunchy...but, not near as math insane as HERO games.
That being said, back in the day, they put out some amazing world books...
If you are new...depending on your flavor of RPG (most of your list is fantasy except shadowrun, fantasy/cyberpunk), I LOVE the FFG Star Wars system. If Star Wars does not float your boat, they are soon to be coming out with a new Generic set of rules called Genesys which will be similar to the Star Wars RPG rules they currently use. I really love this system, it is dynamic and theatrical.