As I understand it bird species listed on 'huntable' quarry lists require close seasons during breeding and/or spring migration periods under EU laws;
I agree, although the argument in the case of woodpigeon is that this is an unnecessary level of protection considering that they are able to breed throughout the year in the UK, that their population in the UK is relatively very high, and the abundance of year round feed.
and regardless of Brexit there are similar requirements under Bern Convention which UK is a signatory to - so having a bird species on the quarry list with no appropriate close season would not be lawful currently.
I disagree.
My understanding, which is very limited, is that the Bern Convention protects the species listed in its Appendices, and the woodpigeon is specifically excluded from Appendix 3. Notwithstanding that, the text of the convention does not actually require a close season, it requires either a close season or other measures to protect population levels as appropriate. A better understanding of what the actual situation is would be beneficial, and perhaps some investigation in that direction would be worthwhile.
However, as per the Shooting Times article linked in the OP, Canada goose is an example of a species with open and close seasons and is also on general licences.
The problem with general licences is, as we've seen, that the law is subject not so much to the ecological requirements, but more to political activism and the political biases of those whose job it is to administer them. While they are appropriate for some species, I really think that protection of any species is unnecessary where it is clearly very abundant over sustained periods of time. UK and EU law is extremely inconsistent in this regard, which whole swathes of flora, fungi and fauna entirely unprotected despite vulnerability and individuals of abundant species given high levels of protection.
The ins and out of all this was in fact all outlined in detail in a Law Commission review of wildlife law for England and Wales before Brexit - but then we had Brexit.
As things stand there has not been a holistic review of wildlife law since 1981 for England and Wales. Reviews have since taken place in NI and Scotland.
A holistic review is well overdue, but I fear it would be unwelcome because of the pressures which would be brought to bear. Many of the organisations, generally described by the media as conservation organisations are highly politicised, biased, and frequently wrong. It is rather sad that the period since 1981 has seen much legal protection, often onerous bordering lunatic of some species, yet the overall result has been one of extreme failure and declines in wildlife overall.
How long the woodpigeon open/close season should be would be a discussion point in the event of a quarry species review.
I think the key issues would be:
1. What months do they breed in? Answer, they have been recorded breeding every month of the year in the UK.
2. Is the population vulnerable to being reduced to a threatened level by shooting? No
3. Are they a pest species? Yes
4. Are they migratory in the UK? Not significantly.
Elements to avoid are the ridiculous stipulations that they should only be shot at the time and place where they are causing damage. This is Corbynesque idiocy - completely bonkers for a species with the mobility of birds, and utterly deranged to have to effectively catch them red-handed.
www.shootinguk.co.uk
open whole can of worms 
Dam difficult bird to shoot 
