> They simply aren't particles the way people think of them
They're not anything the way laymen think of them. They're not classical particles, they're not classical waves, they're not classical anything. They're not really probability waves either, for that matter, or even field configurations - the real object is the algebra of observables. But laymen don't get to dictate the vocabulary of physics, and neither do mathematicians. Fields are the things that would be dual to the observable algebra, were it always equipped with a dual; particles are the things whose creation and annihilation operators would generate it, were there only always such things.
Well, you are still talking about models. The creation and annihilation operators are also mathematical structures. That does not mean, as you seem to imply, that "particle" physics cannot be conceptualized outside of QFT. QFT may be superseded someday and the physical systems we try to describe will still be exactly as they are. I'm not sure if you'll agree with me since I suspect you've got a reason not to.
They're not anything the way laymen think of them. They're not classical particles, they're not classical waves, they're not classical anything. They're not really probability waves either, for that matter, or even field configurations - the real object is the algebra of observables. But laymen don't get to dictate the vocabulary of physics, and neither do mathematicians. Fields are the things that would be dual to the observable algebra, were it always equipped with a dual; particles are the things whose creation and annihilation operators would generate it, were there only always such things.