Professional hat on..
Dog bites need to be examined carefully, what happened before the bite occurred (both immediatley and the longer term past behaviour/experiences).
What happened exactly, in the specific circumstances of the bite.
What happened afterwards.
AND one needs to look at the dog, the management of the dog, the size of the dog, the capabilities of the owner, the home environment, etc etc.
Behaviourists have a 'bite scale' to refer to which helps determine the severity of a bite.
0 - no contact made (since dogs do not miss, a 'muzzle punch' is intentional but not a bite)
all the way through to...
6 - victim dead.
Someone earlier mentioned unprovoked bites - no such thing exists. Dogs have a reason to bite, it may be that the reason is they are ill, in pain, have previously been taught that warning people and communicating is pointless so go straight for the bite, and of course like us they are capable of making mistakes and reading a situation wrong.
The most common situation behind 'unprovoked' bites is actually a dog that has suffered endless provocation, been punished for baring teeth, growling, snapping etc until what you are left with is a dog who bites without warning.
Should a dog be euthanised... depends on whether we know why the dog bit, can we avoid that happening again, can we modify the dogs behaviour or has that gone too far, is the owner willing and able to be responsible, is the physical environment hte dog is kept in suitable, can the dog be found a more suitable home and would it be ethical and kind to attempt any of this if the dog is severely anxious or scared or in pain.
So no it isn't clear cut as to whether a bite = immediate euthanasia nor should it be.
It is sad that people still want to blame and exact revenge on an animal for an action that is almost always the result of human failure.