That is really a distortion of the facts.
Grant drank when he was lonely & depressed. It started in California, yes, but he was not a "drunk". He was a man who DRANK on OCCASION and could not handle his liquor well - he got drunk easily.
The common "Grant was a Drunk!" comments care nothing for the distinction and are designed to imply that he was incompetent, immoral, intemperate, or worse, drunk on the field. That is not true. Read any one of the reputable Grant historians (Simpson, Sampson, Perret, McPherson) and they all agree on this.
I will agree with you that many of his Northern political allies also had a hand in this myth, but it was always necessary for Grant's star to fall in order for Lee's to rise in the years between 1890 and 1930 when the "Lost Cause Myth" hit full stride.
I agree that the Grant story is part of the myth, but I don't see how it can enhance Lee's reputation to have been defeated by a drunk.
Whether a man downs a half flask or a full flask - aside from B/A ratios, when they're drunk, they're drunk.