I think that drinking alone may, and I repeat may, indicate a more advanced stage of the disease.
When drinking is still social and part of an activity that draws us towards people, we are still functioning as a social being. As the disease advances however, driving and engaging with others becomes more difficult. We become more aware that we are not acting "normally." We fear, and perhaps rightly, that others are aware that we are not normal. Our guilt and shame exacerbates this, I think.
The above represents my progression. I rarely drank in public, and avoided activities that would "interfere" with my drinking. I didn't want to smell of alcohol with friends and family, slur words, etc. I could drink and fall asleep whenever I wanted. Leave the empties around until morning.
I am convinced that alcoholism is primarily a social disease. It's most profound effect is the impact it has on our ability to interact and function in society. Relationships of all kinds suffer. It becomes easier to hide than to engage with others.
Just my take. I think the adage is correct. It was with me.
warren