As Mat says the Uk is not beyond reproach when it comes to dog euthenasia. Last year alone the RSPCA "killed" 5633 dogs, the majority of which were healthy but a home could not be found in the short "condemned" timescale the RSPCA put on them (Mainly Bully Breeds).
The US holds an even more shameful record of around 6 MILLION a year, 500 dogs & cats are KILLED each HOUR in Nashville TN alone.
On the subject of "No Kill" shelters, one has to be quite wary of their claims (And I am not tarnishing all with this brush). But a recent survey in the US found that out of 200 "no kill" shelters 90% of them gave many of their dogs that could not be rehomed long term to shelters that have a kill policy.....I'll leave the rest to you
My wife in the last year has helped rehome hundreds of BG dogs into new homes in other countries. I see the emotional strain on her and I ask myself is it worth it, but she continues. Anyone involved in doing this in BG is an angel in my books, and has the strength of an ox.
TNR & Education are the way to go, it has been scientifically proved in numerous countries and even the EU, for all their crap, agree it is the best solution.
As for breeders being the main problem. The figures in the UK speak for themselves. There are literally tens of thousands of Bully breed dogs in uk shelters in the Uk. As there are virtually no street dogs in the Uk then these must have come from owners for some reason or other. Yes some will have come from woops litters, but the vast majority originated from registered breeders and back street breeders to fuel owners who end up dumping them on shelters.
Whereas in BG most dogs in shelters come from the street (and owners). But, anyone dealing in these shelters will tell you, there are more and more "breeder" dogs appearing in the shelters, abandoned by their owners.
Stop the puppy farms, Stop the breeders, TNR every dog, and the problem will resolve itself within a generation or two.
I look forward to the day when dogs microchips are checked at shelters and the initail breeder responsible is fined heavily
Edited to add short tale: In Wales last year the RSPCA were called to a house where the owner died and had 10 German Shepherds. They turned up with a captive bolt gun and killed all 10. Royal Society for the Prevention of Cruelty to Animals....NOT