I have known about QFT for quite some time now and the popular description of what a particle is in QFT is an "excitation in the field". This made me believe that a field is a physical field present throughout space and whenever there is an excitation, we interpret it to be a particle. However, I have taken a QFT course recently and it seems like the concept of a field really has nothing to with particles themselves.
For instance, in a free field theory, a universe with a particle localized at a location $x$ is given by $\phi(x) |0\rangle$. This state is not a particular field configuration or an excitation in the field but instead a functional $\phi(x) |0\rangle:\phi \rightarrow \mathbb{C}$. So this state actually has nothing to do with what our field looks like in physical space. Taking this further, we don't even have a properly defined number operator in an interacting theory although it because of vacuum interactions.
However, when talking about fields, we could consider the "quantum corrected field" given by $\langle\phi\rangle_J = \frac{\delta W}{\delta J}$. Similiar to the physical field above, it seems like this field has nothing to do with particles because it would just be zero in a $\phi^4$ theory or any theory with even number of vertices if $J = 0$. In addition, although I am not familiar with spin, I've heard only scalar theories can have a non-zero expectation value when $J = 0$. It also seems like states aren't even related to the path integral approach as the path integral formulation only seems like a tool to calculate correlation functions.
So in this context, what are people talking about when they say particles are excitations of fields? what field are they talking about? The "classical" field just seems like a generalized coordinate that states or functionals take but doesn't really have any physical interpretation. So what is the physical significance of the "quantum corrected field" and the "classical" field?
