Particles are an approximation to the actual behavior of the field, and are used in perturbation theory to calculate the more complicated field behavior.
This works well when interactions are weak. Electrons do not couple strongly to the electromagnetic field, so it makes sense to view electrons as particles. However, quarks couple very strongly to the strong force (hence the name), so the perturbative approach breaks down, and it makes less sense to view quarks as particles.
So in a non-perturbative QFT calculation which has a well defined particle-number operator, that's just "an approximation" within the theory? What is it approximating?
Also, for context, my question was posed because the idea of "particle number" as well as "quantum states of particles (which are countable) represented in a Fock space" and in general the idea of particles are, like, page 2 of any QFT textbook. It doesn't approximate anything in the theory. Creation and annihilation of particles (and hence the well-defined concept of a particle) is fundamental to the construction of the theory itself, perturbative or not.
Particles are page 2 of any QFT textbook because the free particle is the only system we can exactly solve. In practice, that solution is usually used as the basis for a perturbative expansion.
That doesn't validate your assertion that particles are just "an approximation". Just because it's used in perturbation theory doesn't mean it's exclusive to it.
You're also manifestly wrong on "the free particle is the only system we can exactly solve".
Okay, it's not "the only" system we can exactly solve, but it's 99% of what we solve in practice, and it's the exact solution you'll see over and over again in QFT 1.
The free particle solution is an approximation to reality, because reality includes interactions. There's a mathematical formalism to this that we'd agree on, but you might disagree about how to describe it in words.
This works well when interactions are weak. Electrons do not couple strongly to the electromagnetic field, so it makes sense to view electrons as particles. However, quarks couple very strongly to the strong force (hence the name), so the perturbative approach breaks down, and it makes less sense to view quarks as particles.