That's a tricky question, but Grant didn't have a typical alcoholic pattern. He never drank when anything was going on. No drinking in inappropriate times and places. No drinking under pressure. It was a hard-drinking age, and the stories of Grant's drinking presumably had some basis in fact, but the stories were also all rumor and innuendo, mostly hearsay and mainly spread by people who had personal scores to settle. Grant continued to drink socially throughout his life after the war without any significant incident; he authored a minor literary classic, a model of clarity and precision, as he was dying of cancer; and he had an exemplary marriage and family life. This is not the pattern of a typical drunk.
One of the historians -- I think it may have been Shelby Foote -- concluded that Grant probably got drunk easily and had some accidents along the way, but that he was not a drunk. His Old Army reputation for drinking was based on his period in the Northwest, where by all accounts he terribly missed his wife, fell into what we would today probably diagnose as depression, and self-medicated. The Army was a small, ingrown, and gossipy place, and Grant never outran whatever happened at that time. But that is not the same thing as being alcoholic, which today we undertand as an addiction and view as being chronic and progressive.
Bottom line, if Grant was an alcoholic as we understand the term today, he spent the last 40 years of his life as the greatest dry drunk in U.S. history. But his personal demeanor, family life, and professional bearing don't look like a dry drunk. So I tend to agree with the "sloppy drinker but not a drunk" theory.
P.S. The definitional issue is important here. Most people in the U.S. today drink and are not drunks -- despite the fact that the vast majority of people who drink have an occasional drinking-to-excess episode on their records. If it's with your buddies on a fishing trip and no one falls into the lake and drowns, nothing comes of it. If you kill someone on the highway or humiliate your wife in public or disgrace yourself in front of your boss, it gets more serious. But such an event, however painful, does not necessarily mean you are a drunk. The pattern is the key.
Then history steps in.
Many officers on both sides suffered from serious wounds and continued to serve. Some were hopelessly addicted to opiates as well as alcohol to control the pain. This obvious implications are that senior generals may have been impaired when key tactical decisions were required. There was a suggestion that the union general in charge of the assault at the Battle of the Crater was basically stoned and misdirected the attack. This was probably not the only instance of a command failure due to drug use.