Hi Jessica, glad you found SR, sorry for what brings you here, of course.
As you know, from Al-Anon and your reading, this isn't actually about you or your kids or whether he loves you or not.
He has told you he quit drinking for those long-ish periods of time, but he hides his drinking (until he no longer can), so chances are, he has never had a "dry" period. He lies about his drinking, that's really all you know about it.
So, do you invite him back in to your lives/home? I guess the question is, do you want more of the same life you have been having with him? I can't see that anything has changed at all. He's upset because he isn't at home - but that doesn't fix the alcoholism.
I would be hesitant to have the teens involved in any of the decision making. I commend you for discussing this with them as that should always be an open conversation and they are not little kids, but the decisions are really all yours. If he is asked to leave for a period of time and all his drinking comes to the fore and he goes on a bender, they may inadvertently feel responsible somehow, for agreeing with you that he needed to leave.
Having him back home may seem like the less traumatic choice right now but in a way it just enables him to continue on the way he has been (which is not your problem, it's his).
What do you want? Is more of the same ok or would you prefer he not come home? Is that feasible for you?