For me it is. If social activities have rules, it becomes gamified. I will think strategically in metagame terms instead of just envisioning my character in the fiction.
Role playing and intrigue go hand in hand to me. To me "intrigue" just becomes the equivalent of a board game if there are a transparent set of rules and reactions. That's not necessarily a bad thing, I enjoy board games. But it's a different mindset.
I would also note that "exploring" my character may not be the best word to use. Inhabiting? Assuming the identity of?
Having a set of concrete rules makes the experience less robust and complete to me. There's no limit to the mysteries a DM can think up, no restriction on how we can pursue the resolution of the mystery, no known structure to follow. Just like real life, which to me is far more engaging.
Are you talking about social element around the table or social interaction in the fictional world? In either case I think the rules definitely affect things. In my D&D games it's quite normal for us to interact as our characters. If we're aware of the metagame we'll be far more likely to interact as players of the game.
Some of the most important decisions about game design revolve around what to include and what to leave out. I prefer a freeform approach with guidance and a few tools to fall back on for pretty much everything other than combat.