I wonder if the chestnut weevil is the same one that also attacks acorns? If so, the abundance of weevils may have more to do with neighboring oaks than anything else, especially since chestnuts are about extinct in the wild where you are.
Chickens are too small to eat most acorns, much less chestnuts, so they would have to scratch up the grubs when they exit the nuts and go into the soil (which they will eventually do). So you'd have to pen the
chickens under the trees and just leave them there for several months at least, even after you gather up the rest of the nuts, assuming you do so, so as to scratch the mulch and find the grubs that have already bored out. A pig, or perhaps a ratite or even a turkey, on the other hand, would swallow the whole acorn (and the chestnut too, in the case of the pig), and so bye-bye weevil too.
I've found working with acorns that if I open them as they fall, sometimes the weevils are very small or still just eggs, and if I dry the nutmeats hard in the sun right away, they can be ignored; the visible grubs have not eaten much of the nut and this portion can be clipped away with a pair of hand pruners (this is also how I shell both acorns and chestnuts) But that is for subsistence use, markets will want the chestnuts fresh, and in the shell.
Native peoples often would burn the ground under nut trees of many kinds, which would destroy insects in the ungathered nuts and the mulch layer if done at the right season.