The Burton grip facilitates playing double stops with the same hand. My imagination fails me when I try to find a legitimate use for it on the drum kit. For one, the distances between different sound sources seem to be generally speaking too big for the Burton grip to work.
Steve Gadd used four sticks in his Late In The Evening groove to produce a sound of multiple drummers, but he used the matched grip.
I used to use 2 sticks on the ride cymbal only for trying to replicate the double drummer sound of certain Allman Brothers songs. It was a novelty, but I got good comments from strangers in the audience about it. I thought it sounded interesting, good even. I don't do it anymore though.
Darren King from Mutemath is the only modern day example I can think of who occasionally uses a grip that is similar in the song "Clipping". Check out around 1.05-1.15. Two sticks used to hit both sides of the hats and the rim of the snare