No he doesn't. And you can't convince him. He needs to work it out.
I'd suggest this. But don't do it unless you think and feel it is right, don't do it out of desperation. You can't convince him of anything until he is ready.
One thing BB mentions a lot is how he is finally starting to understand how his actions affect others. He now desperately wants me in his life, but alcohol and his untreated PTSD regularly get him off track. I now try and simply communicate how he makes me feel -- but it's not always that clear. You can't over load them -- they just won't read it for a start.
So this is my suggestion, and please only do this unless your female intuition is saying yes. Only do it if you are completely happy with what you have said. And you need to make it gentle, but final, kinda tough but not manipulative, don't give an ultimatum, simply explain how you feel, how his actions affect you and why you love him. He's asking you to prove he is a good enough, so explain why. But also explain how this all makes you feel. he doesn't want you pity, trust me -- especially as a man, it's the last thing he needs, he feels like a loser already.

Firstly, you need to write it all out, spend a day or two or even three, on and off writing how you feel. Scan the email for guilt tripping, accidentally manipulative comments and delete them -- they won't do any good. There is a fine linne between saying how you feel and how it affects you and telling someone what they are doing wrong -- you cannot tell him what he can and can't do -- you can only tell him how his actions affect you. Keep his kids out of it. It's just about how his actions affect you. Bringing his kids into it will manipulate his feelings and make him feel guilty -- trust me, PTSD sufferers suffer from enough guilt. If he feels guilty for making you feel bad, then that is how it is. He needs to take responsibility for that. Then leave the email for a week, leave any communication with him at bay. Then have a look the week after.

Don't send it for a week because you need to be sure you can stand by what you say. It is important you stand by what you say, if you don't he will unintentionally manipulate you, as he has already managed to begin to do.

In the email, explain all your feelings, email it to me in a private message if you'd like some reassurance about your language. But this email is to end it -- for now. You're gona run out of steam girl -- I am only giving BB the energy and effort because he has proven to me he wants to change and he is not wallowing in bullshit and messing with my head and he's not feeling sorry for himself and playing games. Your guy is playing games with himself and you -- and that is his responsibility -- not yours. I did this when BB and I broke up three years ago. He later admitted it helped a lot. Not straight away, but it eventually sank in. You need to be very clear and detached in this email -- tough love is what you need to express. Then leave it, give him space, a clear break, even if he replies, leave it, because you need to tell him you want no contact for a few months and you don't want to be in contact until he has given therapy a good go and cleans up. You need to be sure you want to do this. And judging from the post you left on my thread, you need to! That's my advice.