I think the GnR could do that pretty well if you push the mids up. I could probably even get that sound with a stock SL2 and a Les Paul if I fiddle around a bit with the EQ and bright/bass switches. To my ears the stock SL2 has more of a midrange hump than the GnR.
With the SL you'd need something on the front end to boost it to that gain first. It's not going make the notes flow like they do in the video by itself. With a good pedal though, the SL is great base for any overdriven sound. Good luck - have fun in your search.