Friends are friends
I don't know as the situation is really as tricky as it may appear at first - if they don't MAKE it that way. Isn't it ironic that THEY are the ones who are living good, happy lives with integrity and feel they are subject to scorn & reprisals from someone who, if the truth were known, might have more to hide than them. Ahhhhhh life is funny !
But the bottom line is that there's nothing unethical about having close friendships - maybe a little closer than many - but TRUE friendships as you mention. The behind closed doors parts are really none of BBs business any more than they would be yours or ours if it were not for this forum. The arrangement could be explained as a "friendly ex", most anything and as long as physical affection is kept under control in public it should be fine. The existence of the kids probably sets the framework for the "traditional" family unit but even that could be explained either as the current or "ex" side of the equation depending on what's most practical. Nothing wrong with "Mom" wanting to spend time with her kids including staying with them for periods regardless of the presence of the - in this case - "ex". Whichever...
There's nothing really so bizarre on the surface to attract attention and strangely enough the truth is very honorable & respectable. If only the other half (2/3? lol) of the population could lead such exemplary lives !
Hope that helps.