In classical mechanics, solving a problem means to give the path of the particle over time, given some initial conditions.

It seemed to make sense to me to define "solving a QM problem" as giving the evolution in time of the wave function, given some initial wave function. And this is what the propagator does.

However, the propagator is the second derivative of the generating functional. In principle, if we think in probabilistic terms (QM is a probabilistic theory), one needs to give all the moments/cumulants of the probability distribution in order to properly define the probability distribution. And in probabilistic terms, DEFINING a probability theory means to give the probability distribution.

As a consequence, giving only the second derivative of the generating functional is not enough to "solve a QM problem", if we understand this problem in probabilistic terms. We should give all the derivatives of the generating functional. In physical terms, it also seems to make sense that higher-order derivatives of the generating functional need to be provided, since there are many-body interactions that cannot be accounted for the 2-point correlation function alone.

In QFT, all this is obvious, and all books work on computing n-point correlation functions. Through the LSZ theorem, we use these n-point correlation functions to calculate S-matrix elements. So, solving a QFT needs all n-point correlation functions, not only the propagator (even though, of course, the propagator is very important).

Also, I note it is not usual to speak about the wave function of QFT. I recall having heard about it, and that in some formalism of QFT there is something called a wave functional. But it is not often books speak about it. If there were a wave functional in QFT, then we could have the same "problem" we have in QM (why don't we define "solving a QFT problem" as giving the propagator only?). But since it seems that this is not the usual language in QFT, everything fits in QFT: what matters are all n-point correlation functions, of which the propagator is only one of them. All n-point correlation functions are needed to define the QFT problem.

But in QM, it seems that with the propagator is enough to solve the QM problem.

Question: why QM is not treated in analogy with QFT, by defining the solution as giving all n-point correlation functions? And if so, why is not enough to give the propagator to solve the QM problem, since the "wave function is all there is", and with the propagator we have enough to completely specify the wave function at all times, given an initial condition?