This issue always falls a little close to home for me, as laying beside me right now are two dogs that, if taken to the shelter tomorrow, would both be euthanized for "behaviour issues". Gaci, my oldest girl, is by all accounts the best dog I have ever had the experience to share my life with. She has taught me more than any other human or animal, she shows no bounds in her devotion (and tenacity...LOL), and is all dog through and through, and I thank doG every day for having her in my life. So it's hard for me to sit and make blanket claims that all _______ dogs should be euthanized.
When Shimmer was six months old she bit my uncle his calf. I don't share that much with anybody, as most people get the wrong impressions when they then meet her. I decided pretty much then and there that she was going to be mine, that I wouldn't put her into the hands of somebody who might make her into a statistic. Because as much as there are degrees to aggression, I consider all of it very important. For public safety, for reputations, but also for the dog's well being and future. Shimmer is turning three this year, has never bitten another person, has not even tried, and enjoys socializing with folks when she warms up to them. To see her become a little ham and suck up to everyone at agility, you would never know that once upon a time she was a certain way.
I have firsthand experience in front of me (and others not in front of me today) to know that dogs can change. They can learn, they can change their emotions, and they can become better members of society.
However, I do not forget that not everybody has the means, or the experience and knowledge, to do as I do with them. Life does change with a special needs dog, not to mention two. For all intents and purposes to meet them on the street walking by you wouldn't really know they had issues - that's how much work we've done. We do competitive agility, we're doing Rally this fall, and we take walks in public places. However, we do not attend dog parks, people do not routinely stop to say hello to them, they don't do on leash meet and greets in the dog park - there are some things that you learn that you just don't need to set them up for. As much as it "looks" normal, there are still issues that would arise if I didn't stay on top of them, and stay alert 100% of the time.
So in terms of a shelter taking in dogs, one needs to realize that the number of people who have the time and energy to work with special needs dogs is not that high, and yes, for a shelter to turn a dog into the hands of the public is always a liability, no matter the behaviour of the dog in the shelter. For the purposes of helping dogs as a whole, I understand why the shelter euthanizes some dogs while allowing others to live - as much as we'd like it, it's not an ideal world and in the end somebody has to dictate which dogs live and which ones do not. Until you walk in those shoes I think it's very hard to judge in any manner. While I could knowingly take on a large problem dog, that doesn't mean that I am about to pass that dog off onto anybody who thinks that dog is cute, nor would I risk the public's safety in that manner.
I hate the thought of putting dogs to death, however I also hate the thought of any dog living its entire life in a 5 x 5 cage for 10-18 years because it's deemed "unadoptable" but yet somebody thought that it shouldn't be euthanized because somewhere, somebody "might" come along that can handle that dog. I will be honest and say I am not a big fan of no-kill shelters because I think it is more cruel to force a dog to exist (not live, but just exist...a dog in a shelter is not "living";) rather than to relieve the unhappiness, loneliness, and stresses of a lifetime behind bars because somebody felt the dog shouldn't be euthanized "just because". And if a dog's behaviour issues are serious enough that the "average" person cannot handle it, then the dog should be humanely euthanized rather than sentenced to a life imprisonment.
It's a sticky situation with no real "right" answer, because just as many great dogs are put to sleep totally based upon their looks or breed biases, rather than behaviour issues, that it's not something I, nor likely anyone, can really make firm claims on, because there's always something that is the "exception".