If the suppression from competition is having an effect, then you’ll have dogs dying in random places from malnourishment. You might also have more aggressive dogs as the hungrier an animal is, the greater risks it will take to feed.
Directly euthanizing them would definitely be more effective. Reality is just too grim for some people to admit.
The populations will just rebound unless you really manage to euthanize all of them. Sterilizing and vaccinating takes care of the problem.
Unless the problem really got out of hand, it's not just "some people" who oppose a total cull. Even if people silently approve of the cull, one will find much less volunteers for such an effort than for a large-scale sterilization and vaccination program.
Directly euthanizing them would definitely be more effective. Reality is just too grim for some people to admit.