I don't see why he should stay. There isn't anything noble about hanging on to this marriage... just because. The kids certainly don't need to be raised in a household where the parents are so cold to each other. I don't understand why he doesn't consider leaving as a viable option, unless he's addicted to being miserable somehow.
As far as continuing to see him, well, you've said that you are only spending time with him as friends now. Okay. Your conscience, and his, will guide your actions. You know people will judge you but only you two can know what is right for you. If I were in your place I might encourage him to examine all the angles, and just look at the possibility of leaving her. Not for you, but for himself. I am sure your friendship will mean a lot to him.
Has your husband met or gotten to know him, as well? He probably could use a good male friend, and male perspective, to help him through making tough decisions.