Put yourself in their place. If I felt my beloved animal was doing just fine and some neighbor took it upon themselves to challenge my ability to care for it, I would be less than friendly toward that neighbor. I'd tell them, MYODB. I can't imagine a situation where I would tell the interloping neighbor "You're right. I'm endangering my beloved pet. It's my pet, and I see nothing wrong with how I'm caring for it, but I'd be willing to give it to you, a virtual stranger just because you think you could do better." In truth, they probably do feel bad, and they probably DO know their pet could be better cared for, but that doesn't mean they'll be responsive to you. I don't think they think you are trying to steal their dog. I think they are considering your offers as attacks on them as human beings.
Not saying you are wrong in calling in the Animal Protection people at all. I'm just saying that you'd have to have a close, trusting relationship with someone to be able to appeal to their better instincts in this situation.
ETA: Here's a similar situation I was once in. We lived in a rural area in upstate NY (kind of upstate). Across the street was a family where the parents worked every day and left the adolescent child at home (she was about 13 at the time). She would come to our house and play with my kids, which was great for me. She was a strong, feisty girl. But one night, when she left our house to go home, she returned quickly, looking incredibly fearful, and said that her parents had arrived home early. Apparently she was not allowed to come to our. house. They viewed me disparagingly as the "brownie-baking mom"--purely a defensive posture. But when we told D that she had to go home, we offered to take her, and she was so tensed up and fearful she was literally digging her heels into the ground. She had told us that when they beat her it was bad enough but when "they got out the hose, that REALLY hurt."
We explained to her parents that we had asked D to come to our house because we really needed someone to watch our kids for an hour, and we were so grateful she was available.
As soon as we got home we called Child Protective Services. I have no idea what happened after that because we moved shortly after. But it was a horribly heart-wrenching experience.
Point being: we would never have won them over or talked them into letting D help us out. We had to call the authorities. We never talked to them again.