We are completely focused on a new Saints Row game. The Third remaster (and all ports/remasters of our older games) are something we are not involved with. Those are a publisher decision and outsourced to other studios.
We are on record that any RF references in other series like SR are just fun easter eggs and not canon/connected. Agents of Mayhem is an alternate universe spinning off one of the choices from Gat out of Hell, though we do not actually consider it a Saints Row game.
Always best to start from the beginning, as it is one story with some continuing characters. Though each game is designed to be able to jump in and get a quick understanding of things without previous knowledge.
As you suspect, we are a supercomputer that has gained sentience and was hired by the studio to answer the exact same questions over and over. How else could we not go insane with such a repetitive task?
Why? The stars somehow aligned with every single decision and event all leading to one glorious (eventual) moment of Saints Row awesome It's going to beautiful.