At risk of overgeneralizing, I'm not sure the GB/wedding scene is where you pick up the kind of "it" you're talking about. I've done some GB work and spent a few summers in a van playing weddings. These gigs in my experience are about providing a variety of recognizable covers of familiar tunes in an entertaining way that allows the audience to basically hear and sing along with the songs they know. Like, they're hearing you, sort of, but you're also triggering their memory of the record and the fun times they had listening to the song before. I haven't generally seen it being about reaching the heart of the music or sinking deep into a groove. You just have to get in the ballpark so they can hear the songs they know and love and preferably be entertaining to look at.
If you want to put a hump in your back and shake your sacroiliac, I would spend some time going deep. Listen to records that embody the kind of "it" that you want to manifest and then digging deep in to capture the feel. Like some here have said, consider not just whether you're faithfully playing the key notes, but the shapes of the notes - the attack, the body, the decay, the dead notes, the little slides and bends, the accented and ghosted notes. Listen to and cop where the notes land on the beat. Are you aggressively on the front of the beat or laying behind? What makes the song feel greasy or aggro? What's the difference between the attitude you get from the original players and the nerd belting out note perfect, soulless covers on YouTube? Feel how the song makes your body want to move. Then try to channel that feeling when you play along or perform. Go see some really good live music - original music, not cover bands. Observe and, more importantly, feel what the band is doing, how the interplay between the bass and drums (and everything else) contributes to that. An oddball exercise that might help is to try and copy environmental sounds like machines or animals, to get out of the rut of just plunking notes you're supposed to play.
To put it another way, use your ego to shift from the superego towards the id. I think the "it" you're looking for lives in the id.