I think it's widely accepted (and I believe) that "drinking + driving + killing someone else = very bad", and that "drinking + assaulting = very bad", but I don't think I know enough about the health implications of drinking.

If a person really was drinking themselves to death of their own free will (i.e. they are aware of the physical health effects, and they do not experience any addictive effects from alcohol), then I don't see an ethical issue with them choosing to do that.

But if suicide rates go up, why could that be?

My opinion is that if someone who chose to drink (i.e. knew the risks and was not addicted) accidentally fell off a bridge or accidentally shot themselves with their own gun, it is their own fault.

When someone is suffering in terms of mental health such that they are addicted to alcohol, or are depressed, or feeling suicidal, then I just don't know. It feels to me like that although the alcohol is not the root cause in these situations it definitely could've pushed them over the edge. I think this is analogous to "someone who felt suicidal committed suicide after another person told them to jump off a building"... I really don't know where I stand on the issue.