You can also watch this clip of the Vantage performance in which Simon discusses his upbringing in playing jazz in his dad's band in England. He does
have a history in jazz and wide, deep set of experience in playing styles, just different from many of the American masters (Philly Joe, Shadow, Roy, Elvin, etc.):
Obviously, Simon can play jazz, and this piece demonstrates that he can play it quite well and with some flair, as, I think, Steamer noted.
His kind of jazz playing reminds me of Bill Bruford's in Earthworks: very good, very accomplished, but different than players who immerse themselves fully in the tradition (e.g., when you listen to Miles, Coltrane, Monk, Herbie, Joshua Redman, etc., and hear the drummers there). As others have noted, Simon (and Bill) has extensive experience as a rock/fusion drummer, even though he (and Bill) started with jazz and had jazz as one of his first inspirations.
I think what's really at heart here is immersion. From my experience in teaching and research (in humanities, not music), if you want to really learn something well and do it well, then immersion is necessary--at least for most of us. Whether it is literary study, science, math, philosophy, a style of music, I think you must put in your time to inhabit the space of your field of study/interest. The more you do this, the better your chance at getting a deeper knowledge of your subject, and the easier it becomes for you to call upon such skills when needed.
Players like Blakey, Elvin, Haynes and most other greats mainly spent their entire careers mastering jazz and forging new directions in jazz. They were not playing rock sessions or fusion sessions or pop sessions. Not even Tony's hybrid Lifetime stuff is really rock, in my opinion; it is still heavily jazz oriented. A player like Simon has a bit of the Renaissance man about him, in a way: he can play a massive variety of styles extremely well, but by doing this perhaps he sacrifices time and focus. He is a solid all-around player, but, because of this focus, he will probably always have a hard time playing jazz at a deeper or more sophisticated level. This is not a criticism of him or his playing--it is just a simple observation of where a player has chosen to invest her or his time and energy. You can have a jazz player who has spent 40 years in jazz, while Simon has spent equal time mainly playing rock/fusion with some jazz and other styles mixed in. Will the jazz player sound different when it comes to jazz? Yes, because in this example the jazz player has a lot more focused experienced in a style. Whether people want to label that difference as "authentic," etc., is up to them, I suppose.
Note: I will qualify my immersion argument by noting that if you don't learn how to study well and smartly, then there is a good chance that progression will not happen. Immersion, by itself, doesn't guarantee growth and mastery. Hence why we all benefit from good teachers (Alan Dawson, Ed Soph, Gary Chaffe, etc.)!
As for the point about the sound of Simon's playing, I think that has more to do with the fact that he's got clear heads on his Starclassic Maples and doesn't appear to have his bass or toms tuned to the usual high bop pitch. Tune up a bit and put coated heads on the drums, and they'll sound different.
I will add, as a side note, I'm glad to see Eric Harland's name mentioned in this thread. He is a phenomenal jazz player. I think he and Matt Wilson represent some of the best playing in jazz today: steeped in history but pushing forward at the same time, with fun experimentation.