If we're looking at it purely from a statistical point of view, then yes, the numbers might not look that good in favour of "more drinking + ridesharing is better than less drinking + drink driving".
Ethically, I consider people drinking themselves to an early grave of their own free will to be "less bad" than a drunk driver killing someone else on the road or a drunk person committing a violent crime against another person.
I make this point in the context of the rest of what you've said, though, which raises good discussion points about "should we fix X really bad thing if it leads to people doing more of Y maybe-less-bad thing?"
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.
Taxis (at least in Melbourne) used to be uniquely awful for getting home after a night out. They'd make you "prepay" a ridiculous amount like $50 cash for a trip that should have cost $15 and then refuse to give any change at the end of the trip. For the specific set of customers I was in, taxis were more expensive than themselves because they were basically stealing from us.
10% seeming smaller than 62% is not a fact that anyone should be focusing on.
There are actual numbers that can be compared in context; the fact that a percentage of one thing is expressed as a larger number than the percentage of another thing doesn't mean it's more important. Very few people die of drunk driving accidents (as compared to the number of people who drink), so if 20 fewer people in an area per year die from drunk driving, but incidents of binge drinking in that area increase from 1 million to 1.1 million, it's not simple to say that the outcome has been positive. Maybe 21 more people get esophageal cancer. Maybe 21 more cases of domestic violence end in a homicide.
That being said, this is a wealth effect and inevitable if you're making progress. People are drinking (out) more because their perception is that with rideshares it has become safer (easier to get home.) A cure for HIV would cause an increase in the spread of other sexually transmitted diseases, for example, but that doesn't mean that you shouldn't cure HIV.
One material difference I can see between rideshares and a cab is that rideshares are often cheaper because 1) they are subsidized by investors and 2) they can be shared. This doesn't say anything interesting about Uber; if you wanted to make cabs cheaper, you can tax them less or subsidize them. The other, which I think might be more important, is that many rideshare users don't feel the need to buy a car, and that the rideshare demographic is largely similar to the binge drinking at bars/clubs demographic. Of course, taxing cars or the parking around bars/clubs could have the same effect.