I’m also going to throw a spanner in the works here. Life is not black and white, living together does not necessarily equal together.
I am separated from my husband, but we still live together. Our situation is very much as you describe your partners situation. We have secondary school age children.
We all live in the same house. We are separated but not divorced. We sleep in separate rooms and separate lives. We most definitely do not have sex! But we parent together- we go on family holidays together and celebrate family events together.
We both have relationships with other people. These are invariably complicated by the situation and the fact that currently we don’t want to tell the kids about other partners- because neither of us has plans to live out currently.
Yes, we could get divorced and one day we probably will. But it’s bloody expensive and neither of us want to see the kids only 50% of the time. Living with your ex partner makes it much easier to go away- for work or to see other people.
He knows about my new partner. My kids do not.
I think these arrangements will become increasingly common as our crazy housing market makes it so hard to divorce and start again. We simply could not afford two houses in this area and we aren’t disrupting our kids education to move.
You of course, don’t have to accept a partner in this situation. My point is simply that not all men (or women) in this situation are cheating on oblivious partners.
FWIW if a woman asked to meet me to see if my situation was genuine then I would.