For those leaping to burn the OP at the stake here...
This is someone who has come to ask for help in rehoming their dog.
The minute someone doesn't want their dog, for whatever reason at all - that dog is better off rehomed. No dog should live where they are not wanted.
So the reasons are in fact irrelevant.
What are you achieving by attempting to make the OP feel awful, because it isn't going to make them keep their dog.
It is likely to make them not ask for help any further, and as they have dogs they need to rehome, who do you think suffers as a result of that?
I am also firmly in the 'there are worse things to happen to a dog than euthanasia'. Much much worse things.
It's nice and easy to say 'oh you're a horrific person for even thinking it' but have you been the person living with dogs you cannot keep for some reason?
Have you had to leave your dogs to suffer separation anxiety because you have to work, or struggle with dogs with behavioural problems. Or have a timer running on how long you can keep them because you're about to be homeless... or have to decide if you buy dog food or food for the kids.
I know lots will pipe up and say they'd live in their car for their dogs or they'd go homeless to keep their dogs, but the truth is most people not only wouldn't, but couldn't.
Of course, there may be other options. There are fostering schemes, there are rescues and people who foster dogs in domestic abuse situations. You could potentially kennel dogs whilst you find a suitable home for you all...
But if people are shamed for asking for help... they won't find these options!