You may not view Peart as important, but the annals of rock history do. Remember that the premise of this thread is best ROCK drummer, not jazz or anything like that. Countless rock drummers over the years have called Peart a big influence, and it is kind of silly to say otherwise.
In reality, most rock drummers called Ringo and Bonham influential than Peart. Billy Cobham was a bigger influence than Peart. I never hear Steve Smith talk about Peart as much as he does Cobham. If you're lumping in Portnoy and "progressive" (see: DT nuts) drummers, then I still don't hear Peart in any jam band drummer, who have chops but play them in a different way.
The bottom line is you don't hear Peart's influence in rock music, other than listening to again, DT. Dream Theater itself isn't that influential in rock either. Bill Ward of Black Sabbath could be a bigger influence now than Peart. A guy I know in Brant Bjork has influenced more drummers with his brand of Stoner Rock drumming, which is huge in Europe now. He hardly plays drums anymore, lol. Modern Drummer called him "The Influence". A lot of these rock drummers are post punk drummers, who also borrow a lot from Bill Ward.
The town I live in - where Kyuss and Queens of the Stone Age are from - almost every rock drummer in town has this kit: one 26" to 28" bass drum, single rack tom, and one to two floor toms, with a few cymbals.That's it. It's Bonham, Ward, and yes, Bjork who they are sleeping on. You go to Coachella and you don't see a huge Peart kit onstage with most of those bands. It's the single rack tom variety. My old huge Phillips kit would meet derision, lol.
Funny story: a buddy of mine plays in one of these stoner rock bands (I know a few that do), and I had an old steel concert bass drum, that I think was 30" in diameter. I told my buddy I would give it to him once I find it, because I thought I put it in storage, and hasn't seen it in years since I moved back in town from SF. He called me almost every day to see if I did find it. I didn't, much to our dismay. I think one of my nephews got a hold of it and gave it away. But the funny part is the bigger the bass drum the better. That's how crazy these drummers are here.
All of which shows hardly a Peart influence.
I dunno how anybody can think that jazz does not influence rock, or that it's irrelevant to this discussion. Watts, Baker, Mitchell, and even Ringo were influenced by jazz drummers early in their careers. It's irrelevant to
your point because it doesn't fit into the Peart narrative. Yet, Burning for Buddy down? That's Peart's pet project. Trad grip down? Peart was re-engineering his technique for it. He is going back to his jazz influences, so jazz vocabulary is very relevant to this discussion.
I dunno if I already posted this, but this is where jazz vocabulary and rock meet:
https://www.youtube.com/watch?v=jA-O2ylCHd8
Carlos Santana and Michael Shrieve were knocked out by this band, as well as a lot of other musicians (even Jimi Hendrix) when they came out. Terry Bozzio, Steve Smith, and Vinnie Colauita all credit Williams as a huge influence as well.
Speaking of when jazz vocabulary meets rock:
https://www.youtube.com/watch?v=xBG6IaSQCpU
Yes, the famous Woodstock clip with Michael Shrieve, whose hero was Elvin Jones. Yes, he was peaking on acid with Carlos as well.
You're trying way to hard to pin Peart as the greatest rock drummer ever, while not trying hard enough. He simply isn't.