I think these things usually come down to perceptions of human intervention as basically bad. If you actually witnessed the death of every individual dog in your scenario as compared with euthanasia, it's not clear that it would be more pleasant at all. These dogs will die diseased and malnourished by the roadside. Again, I am not really at odds with your emotional position, I'm just interested in the moral foundations on which we sit, if that makes sense at all.
If I may make an attempt at the moral underpinning, it is something like "survival as intrinsic worth". There are flaws to this position, as with any.
Setting aside the fact that I'd be witness to less deaths because I've chosen a more effective protocol for population control, I think letting them live out their lives isn't a net negative. Why would I expend energy ending lives if they don't impact mine?
Granted, there are times when euthanasia would be necessary. For example, feral dog populations in areas where the vulture population has died off, where they carry disease and attack passersby and the supply of food is basically uncontrollable, I get it. But that's a case I personally would preemptively qualify if I were considering it.
In countries with well-managed stray dog populations, people actually feed them, and they get rid of carcasses and other leftovers on the street. Their life might be harsh, but not entirely unpleasant. In countries with unmanaged populations, people will be harsh towards them and mistreat them. And they will also be more aggressive towards humans in return.
If I may make an attempt at the moral underpinning, it is something like "survival as intrinsic worth". There are flaws to this position, as with any.