It is true that the Self (ātman) cannot be a drunkard, for your true nature is one with Brahman. But if a person is drinking alcohol to excess, then surely he hasn't yet realized this true nature.
The Bhagavad Gītā says,
BG 3.17: He whose delight is only in the Self, whose satisfaction is in the Self, and whose contentment is only in the Self, for him the need to act does not exist.
BG 5.24: He who finds his happiness within, his delight within, and his light within, this yogin attains re-union with Brahman, becoming Brahman.
So it would follow that, he whose delight is in intoxicants, whose satisfaction is in intoxicants, or whose contentment is in intoxicants, for him there is work to do. If one has not become absorbed into Brahman, then he must still be inflicting himself with māyā. And, how can we remove māyā if we do not first admit its source?
BG 6.27: The yogin whose mind is perfectly peaceful, attains the highest bliss; he whose rajas has been subdued attains re-union with Brahman and is free from erring.
BG 16.23: He who—by acting under the impulse of desire—discards the scriptural injunctions, attains neither perfection nor happiness nor the supreme goal.
To subdue our rajas (passions), we must admit what they are. To stop acting under the impulse of desire, we must know that we act so. We cannot eliminate ignorance with ignorance, only knowledge can do so. Thus, the alcoholic must first recognize his actions and how they effect himself and others around him in order to eliminate the obstacle.