The elephant (literally and figuratively) in the living room is his drinking.
It bothers you. I glossed over the the snoring on the sofa issue, the kids not being fed etc because bottom line is you know that his inability to be a full-time partner is due to his drinking and whatever other psychiatric issues he has. It is a full deck of cards you and your kids are dealing with.
That is where you have to really ask yourself what it is you need and want. Reality is that drinking is progressive (as someone pointed out). My impression is you want a sober partner, which you do not have. Normalizing what he is doing doesn't get him to recovery sadly.
Al-anon would really help you and understanding alcoholism may also give you a 'bigger picture' of his disease. (Under the Influence, by Milam and Ketcham, is very helpful and informative).
I guess for me, I could treat the alcoholic like a lamp in the corner but it still doesn't change the fact that he is slowly poisoning himself. Would be hard for me personally to witness so remember that your kids are also experiencing this and being affected in long term ways.
Just some things that came to me while reading your post
