The Foo Fighters are something different, agree, though their love for Rush is acknowledged. I do feel the Foos are the heir apparent to the hard rock throne. They are the number one band in terms of still making hard rock music, getting mainstream airplay, being the top ticket. They are the heir apparent to U2, AC/DC, Guns n Roses and Metallica as "America's biggest hard rock band". Their Rush covers aren't bad of course, but they suffer vocally (as does just about every Rush cover). Still, their exuberance, their love of playing and literally having open ended shows like they do make them heroes in my mind. They play every show until they are tired, sometimes yielding near 4 hour shows with covers, extended jams etc. No encore ever, just play until they drop.
From a purely nuts and bolts perspective, as a musician, I disagree. I've yet to meet a (serious) musician (not a teenage boy) who cites The Foos as their musical influence. Even with a younger crowd, you still see Rush and many other bands referenced. In 20 years I suspect Rush will still be on people's minds in the music world, and the fickle casual rock fan with have moved on to a couple dozen other Foo-like replacements.
There's a reason the Foos cover Rush - they are the students and Rush is their teacher. Showing respect and admiration does not make them peers/equals.
There's a lot of shitty bands that draw big crowds. I don't consider touring success a measure of greatness. Metallica still draws and they are are far from their prime at this point. if we were to measure greatness on a live show scale, Dream Theater and the complexity of their music pulled off live would take the crown in this Rush-like discussion. But like Devin, they're mostly metal in a way Rush mostly was not, which is why I wouldn't consider them heir apparent. Influenced by, but not Rush 2.0