The fact is, Adventure Sonic- despite the name- was never restricted to just using stuff from the Adventure games. The far thok, being based directly on the airdash in the Advance games, was there from the start, not to mention the whole Advance game flair a few versions later. And here's some fun facts for ya- Sonic Rush released in late 2005, believe it or not, a full year before Sonic 06, 3 years before Unleashed, and a year and a half after Sonic Advance 3. It's closer to what people consider Adventure era games in both temporal directions than it is to the first console boost game.
But, honestly, that's all semantics if you ask me. Probable fact is, MetalWario and Golden Shine thought Rush Mode would be cool and fun, so they did the work and added it. Enough said. It really doesn't matter whether Rush is an "adventure era game" or a "boost game" in the face of that. On top of that, internally, Adventure Sonic is considered to be "Yuji Uekawa Sonic", inspired by the man's art foremost- and wouldn't you know it, that man's art is one of the common denominators linking Adventure, Advance, and Rush.
Really though, I think all anyone should be concerned about is "Oh wow, MetalWario and Golden Shine gave Adventure Sonic a whole new play style! That's so cool! That rules! I'm gonna have so much fun with it!", not all this semantic nonsense. Just play with the dang thing.