The old spartan soldiers were said to having to do the same thing. Bring up a dog and kill it at the end of their training. The same practice can be seen replicated in Game of Thrones (I'm not sure if it actually was talked about in the show). The unsullied did the dog thing too.

Even though the kill-your-dog-idea works independent of knowing the historical facts, it traces back to the very real idea of how to train hard, mercyless soldiers.

Also, remember that it was higher-ups that told them to shoot the dogs. They are supposed to be super soldiers/spies/agents, and should maybe just work on a need-to-know basis and listen to their superiors.

Furthermore, maybe the old English ways simply value dog lives a lot less that any human life, and to the older knights, this didn't seem as cruel as it did to modern, secular Eggsy