Why we cannot integrate out the light field/modes in effective field theory?

The purpose of and EFT is to compute observable quantities when you are in a situation where some degrees of freedom are hidden since you are probing the system with a sensitivity that is not enough to resolve the structure.
In particle physics, generally this means that you don't have enough energy to produce directly a particle. However its effects may enter indirectly (as virtual particles in some processes), and that's why you look for effective operators in a $1/M^n$ expansion, "integrating out" the heavy field.
Then you see that physically it does not make sense to integrate out the light dof, since you can directly see the light particles and their effects are not hidden and suppressed (in fact, you observe them!).
Mathematically it does not make sense either since the expansion in $1/M$ would not converge ($M$ must be big) or, in a borderline case, the correction to the prediction given at the order $n$ is comparable to the one at the order $n+1$, meaning you should sum all the corrections. This is more or less equivalent to not making the expansion, ie not integrating out the light dof.

The point of EFT is to produce local (effective) actions that describe the low-energy/long-distance physics. If the mode integrated out is of mass $m$, the action of the remaining mode will look local as long as the energies (resp. distance) are small (resp. large) compared to $m$ (resp. $m^{-1}$).

Mathematically, one can, of course, integrate out the light mode. The problem is then that the effective action of the heavy mode (of mass $M$) will be non-local for the typical energy of the heavy mode (which is $Mgg m$). In theory, one could use a non-local action to do any calculation, but in practice, it is really not practical (typically, perturbation theory will fail). This is even worse for massless modes.

The notion that effective field theory only considers "integrate out the heavy field/modes" is false.

Case in point: QCD. You can regard the perturbative (asymptotically free) QCD as "integrate out the light field/modes instead", since the low-energy effects (e.g. confinement) are "integrated out".

As an effective field theory treatment of QCD, according to QCD sum rules (or Shifman–Vainshtein–Zakharov sum rules), long-distance (light modes) quark-gluon interactions are parametrized in terms of universal vacuum condensates or light-cone distribution amplitudes (see here). Therefore Shifman–Vainshtein–Zakharov sum rules can be regarded as an effective field theory which "integrate out the light field/modes instead" and represents the light field/modes in the parametrization of the operator product expansions in gauge invariant operators.

The asymptotically free QCD concerns an energy level higher than Shifman–Vainshtein–Zakharov. Thus the "light field/modes" parametrized in the effective SVZ model are entirely "integrated out". And you are left with a pure Yang-Mills type asymptotically free QCD. Therefore, at high energy level, you could safely disregard the non-YM Lagrangian terms involving "light field/modes" which would surely come into play around or below $Lambda_{QCD}$.