The problem with that assumption, is that it looks at the situation in isolation, for me its a - One pair, square area how much impact on songbirds, finger in the air guess.
From my house I have/had 5 nesting pairs of Magpies that are meticulously picking away at the songbirds nests, or they once were.
The guidance allows them to be culled and directly relating to songbird protection. These guidelines are there to be adhered to and followed, of course, but if this was say a rat infestation and predation of the same with say, badgers, or foxes, we would be discussing this right now as an active campaign by DEFRA to eradicate these animals. Magpies/Corvids don't seem to have that impact.
I get the save the hedgerow campaigns and support them, however, with intensive farming being the root cause and up against big business, the Linnet, the Dunnock, The Bullfinch, sadly doesn't stand a chance.
I think we are largely on the same page here.
I skimmed a couple of scientific papers, a report on a meta study and a PhD thesis and while I would not claim that my search was in anyway comprehensive, the rules are based on a bit more than an assumption. The BTO study was from 2010 but was based on 40 years of data from 200 sites (
https://www.bto.org/about-bto/press-releases/are-predators-blame-songbird-declines).
The evidence on corvid pressure on
overall populations of songbirds is currently weak which is why they are not considered a pest species or 'vermin' like rats. This seems to me to be a good thing as is stops the 'open season', wholesale persecution of a species without just cause.
Where there is a specific threat to birds on red and amber lists, as you say is the case on your own patch, the general licence GL40 may well apply and would permit control by prescribed means, but anyone carrying out such actions should be clear about what is allowed and how they might evidence compliance if challenged.
I raised this partly because I think there is a risk of demonising species uneccessarily (pursuing ill-conceived policies will not aid the wildlife), but also because I wouldn't want any of the bird lovers in this thread to find themselves in trouble because they assumed that killing crows and magpies is allowed in any circumstance. Chasing them out of the garden with foul language and water pistols is one thing (I have done this myself), but I think it is worth checking carefully what the law allows before considering trapping, shooting, destroying nests etc.