I'm not sure I even accept the idea that strays should be euthanized. It seems to be predicated on the notion that strays will necessarily suffer because they aren't under the constant care and supervision of a loving human. But is even that reasonable?
In the general case it may be, but for any specific animal, I can envision quite a decent life in an urban setting. Norfolk Virginia, for example, doesn't get impossibly cold in the winter (average low temp around 33 deg at worst), and people may be quite willing to offer food/water. But even in those cases where an animal isn't doing well on its own, that ought to be obvious to a trained veterinarian instead of assumed.
What is the case for killing a healthy animal who is out on its own? It must be based on some predicted and feared outcome and used as a preventative. But it is odd to think that a healthy stray should die "for the good of dog-kind." If dogs are packed-up and attacking, that's a good case, but sans a valid reason, why don't they do the catch-neuter-and-release they might do for cats?
I know one justification for this round up was livestock predation, but then the chihuahua would be hands off. "Kill 'em all and let God sort them out" doesn't seem fair. It sounds like PETA supports animal rights, except for the rights to life, liberty and the pursuit of happiness. Because, if you want to see a happy dog, watch one running free.