I’m a professional ecologist, though mammals aren’t my speciality.
There is no published evidence that I am aware of that grey squirrels are driving observed declines of woodland birds – see here for a decent paper on the subject.
Some questions for you OP:
How do you know that if the grey squirrels didn’t predate the nests of birds that you are apparently trying to protect because you killed them, that survival rates of their chicks would increase?
If there are too many birds (including the ones that you are trying to protect), they may not have sufficient food, so their young might not fledge anyway. So even if you kill everything that is apparently killing the chicks of the birds you are trying to protect, there’s no guarantee you increase their survival rates. Had you considered this?
As other people have noted, and I agree, cats are very likely to be a much bigger issue for killing birds, including the ones you’re apparently trying to protect by killing squirrels. Are you going to kill all of the cats too? Other birds will also predate them. Are you going to kill them?
If you’re not going to kill all the other potential predators, why are you specifically targeting the squirrels? Do you think that if the squirrels are eliminated, other predators might take more chicks of the birds you are trying to protect? I think there is a reasonable chance.
In any case, I can’t see a reason why more squirrels won’t just move in to replace the ones you kill, if the habitat is good. So are you going to just carry on killing squirrels forever? If so, why?
A final thought – if you had red squirrels, they would also be predating bird nests. Would you cull them too?
In my view you should only use culling if there is a clear benefit. In this case I do not see that there is. I think that just because its legal to kill squirrels doesn’t mean you should. It does not look like this has been well thought out, and I think you should strongly consider removing the trap until such time as the benefit to killing squirrels is clear – which may be never.
Sorry for the long post but I think most of it is important.