I think, in general, if the dog can't have a good quality of life most of the time - and especially inside its own home where it should feel safest and be most comfortable and predictable - through tools, techniques, and management that are feasible for the owner to implement without severely damaging their own quality of life (eg reasonably affordable, doesn't prevent the owner from being able to work or attend to their own or their family's needs, doesn't become an emotional burden that triggers serious mental health concerns, etc.), OR the dog is a danger to other (especially domestic) animals and/or humans in a way the owner cannot or will not manage, AND rehoming is either unfair to the dog (putting an anxious dog through a shelter, for example), potentially dangerous to the new owner, or straight up nearly impossible because the dogs' needs are so extreme, euthanasia is reasonable.
Of course this is deliberately vague, because so many factors are in play. Some owners will take a serious anxiety case and be able to put the money, time, and emotional/mental labor in and make it through, and others are not willing or able to do that, and that's okay. Some communities may be hotspots for Dog People with more homes than average willing and able to take on project dogs and therefore rehoming becomes an option in more scenarios, whereas in other communities anything more than very basic behavior issues are going to struggle to find a home and sometimes euthanasia is kinder than sending a stressed or fearful dog through multiple unsuitable homes or sitting in a shelter system indefinitely.
I personally feel that serious, level 4 or above biting incidents without major extenuating circumstances (eg dog was being tormented, was severely ill or reacting to medication, etc) are grounds for euthanasia in most circumstances, as there's relatively few homes that have the experience (or money to hire experience) and ability to consistently and securely manage such dogs. Below that, I could see situations where almost every genuine behavior problem (eg stuff that's beyond a dog being a dog) might be reasonable grounds for euthanasia in certain circumstances.
Of course this is deliberately vague, because so many factors are in play. Some owners will take a serious anxiety case and be able to put the money, time, and emotional/mental labor in and make it through, and others are not willing or able to do that, and that's okay. Some communities may be hotspots for Dog People with more homes than average willing and able to take on project dogs and therefore rehoming becomes an option in more scenarios, whereas in other communities anything more than very basic behavior issues are going to struggle to find a home and sometimes euthanasia is kinder than sending a stressed or fearful dog through multiple unsuitable homes or sitting in a shelter system indefinitely.
I personally feel that serious, level 4 or above biting incidents without major extenuating circumstances (eg dog was being tormented, was severely ill or reacting to medication, etc) are grounds for euthanasia in most circumstances, as there's relatively few homes that have the experience (or money to hire experience) and ability to consistently and securely manage such dogs. Below that, I could see situations where almost every genuine behavior problem (eg stuff that's beyond a dog being a dog) might be reasonable grounds for euthanasia in certain circumstances.