A classical field $$\phi(x) =\int\frac{d^3p}{\sqrt{E_p}} (a_pe^{-ipx} + a^*_pe^{ipx})$$ is not a unique thing. We could "choose" $a_p$ and $a^*_p$ conveniently so that it represent field of our specific problem.
But when we quantize it we get
$$\phi(x) =\int\frac{d^3p}{\sqrt{E_p}} (a_pe^{-ipx} + a^{\dagger}_pe^{ipx})$$ then $a_p$ and $a^{\dagger}_p$ are unique thing (ladder operators of a SHO). So who this field represent our specific quantum system?! We are in Heisenberg picture so we expect dynamics in operators but rigidity of $a_p$ and $a^{\dagger}_p$ don't let any dynamic even for interacting fields. What is the overall picture of QFT in this manner?