This reminds me of the situation of a friend of a very good friend. His situation looks a lot like your husband's.
Bottomline his wife cheated on him, they have two young children, he tells her and close friends that he is single, and that she needs to move out from the marital house. He is staying married, and he still demonstrates affection to his wife. I wouldn't know if they still have sex, but it just sounds obvious. He freely admits to pursuing casual encounters and has no intention to care if someone would rather he doesn't. He still invites her to public events and he doesn't advertise being single here.
When he explains himself more in-depth, that's how it goes: first, yes his wife is demoted to FWB as far as his emotions are concerned. If she wants to remove the benefits she's free to do that, he knows that's a possibility and he is fine with it.
He says he is single because he is not investing commitment to his wife, not romantically vulnerable to her, and there is nobody he is in a romantic relationship with. Which means he is free to meet up with whoever he wants for whatever he wants as long as the persons are okay with it.
He stays married, because the marriage contract is a helpful tool for them both to devote themselves as a team to their one common goal left: their children and living with them. Protected stable finances and mutual next of kin if necessary. (His wife is nonfunctioning without someone to save her finances. She doesn't acknowledge it but she's the only one. It could escalate to her being unable to be there for their children.)
He says he invites her to events as their daughters will need their mother being part of events in their lives. As for not advertizing there that he is single, he has no intention to pursue anything at such events, so it wouldn't serve that purpose. The only purpose it would serve then, is bringing drama to these events. People say they hate drama but we all know the truth and how morbid curiosity is what they would all be after. He estimates there is no reason to bring this to these events.
One key difference though, is that he never betrayed his wife before she betrayed him, nor did he have any interest in pursuing other people with or without permission. She would accuse him of having had an emotional affair with his,childhood best friend and interpret his acknowledgement that they had deep conversations as an admission of the affair. But there was no such thing. She is delusional on many fronts.
