But back to the idea of shutting down threats of estrangement. As I said earlier, I don't like threats. And I agree that it's wrong for AC/CIL to throw this threat at the GPs whenever they're trying to get something they're not entitled to, such as money or childminding. Or if they say it over "every little thing" just as a way to shut the GPs up, etc. But if they're saying, in effect, "if you don't follow our rules for our children" or "if you don't stop trying to get us to change our lifestyle," or point to some other repeated offense, then IMO, that's a somewhat different story.
Maybe they still shouldn't threaten. But my concern is that if the GPs ignore the complaint and just focus on shutting down the threat, the AC/CIL may feel that's it, they're done, and just cut the GPs off. (NOT saying that this was the situation in your case, Mebster, just speaking in general). For example, if the AC says, "If you tell us what to do one more time, we're done, you'll never see these GC again!" that's awful! But if the GPs respond to the threat only, as in, "Don't threaten us!" the AC may feel the GPs have not heard the message that came w/ the threat, the concern about the GPs trying to run their lives or whatever. And so, the AC might just stop threatening and make the CO. For this reason, I would be very careful about how I shut down the threat and make sure to acknowledge the complaint and promise to change the offending behavior if I could, etc.