Good question - first off, I am starting from a different point with my chldren as I do not have a long history of hurt and damaging treatment of my kids, but am in a position of love and trust with them, which is a help.
Second I have had a long period on my own (my xh goes from woman to woman with no break)and they respect the work I have done on me, and the support I have given them. We were an exceptionally close and loving family, and my xh's behaviour and treatment of both me and them during all of this shocked and hurt them deeply
Third I have good taste (the kids do not like this woman)
Fourth, at my age, my family matter more than a fledgling r with a man. Perhaps that is a male female difference?
Fifth, my kids see me as a responsible adult capable of making good life choices 0 they see their father as someone essentially out of control with appalling taste in women
So if I met someone I would invite the children to meet him either in a restaurant, or social gathering or at my home, and if they didn't want to I would not insist. I certainly would never show up unannounced on their doorstep with new man in tow.
If they met him and didn't like him I am afraid it would be curtains for the r - they are adults with very good taste in people. They want me to be happy and are not opposed to either of us having a relationship
They expect their father to treat them with respect for them and their homes. They will meet OW at his, or in a restaurant, and if he asked if she could come when he sees them in their homes, it gives them a chance to say they do not care for her. He just assumes that they must like her and they must have her in their homes, without even having met her in the first instance.
Do you show up at people's houses with an uninvited guest in tow? I do hope not.