What is the outcome you are hoping for? None of us can tell you anything with any degree of certainty, and only you can decide when someone else's drinking is too much of a problem for YOU to live with.
Based solely on what you've described, he has verbally acknowledged that he drinks too much, but as far as his actions are concerned, it appears he is saying that he has no intention of actually changing anything.
Living in worry, walking on eggshells -- that is not a path to fulfillment in a relationship. If he is an alcoholic, you should know that it is a progressive condition and unaddressed, will get worse. "Functioning" is not a TYPE of alcoholism, it is merely a STAGE.
You know, it is enough if this is a problem for YOU, whether or not he acknowledges it as a problem for HIM.