I think the idea is that if people didn't have access to alcohol, many of them would realize they have no good way to calm their nerves, and therefore realize they have anxiety issues.

I know, and I'm saying that's more so the case for marijuana than alcohol. I've been smoking weed recreationally since college and still enjoy doing so, but the only time Ive done it more than once a week for an extended period of time was during an inordinately stressful time at work a few years ago (right before I ended up quitting and taking a sabbatical to travel). Alcohol is ubiquitous and easy to get, so it's by far the most common for self-medicating many things, but in the hypothetical we're talking about availability isn't relevant.