I think it's true in the right-width-of-sense. Illnesses like depression are difficult because they certainly _do_ lie somewhere closer to the symptom side of the "symptom-disease" spectrum than cancer.
Alcoholism is good for comparison, because it presumably falls even further towards the "symptom" side of the spectrum. But there is certainly a spectrum!
So we have Cancer which is mostly-disease and Alcoholism which is mostly-symptom, and Depression which is somewhere in between.
The fact that these complications lead to weird moral baggage (socially-derived feelings of guilt, etc.) is completely separate from a good analysis of the situation. Depression and cancer are just very different phenomena.
There's as many known external factors for cancer as depression. And as with cancer, they've got only a small impact of likelihood of reducing your risks of getting the disease.
It's therefore unknown if depression is caused by external factors, or simply triggered by them, or a combination, and if so, we're not exactly sure what all the factors which have an influence are.
As it stands today, it's still very possible depression is caused by generic defects, bacterial infection, virus, or some other unknown external factor like polution, exposure to some toxins, or very early development problems, etc.
Honestly, I'd love to be enlightened on some very definite research that knows what causes depression, but as far as I'm aware, science doesn't know.
So there is this problem that some people have depression, and we assume it's just there lifestyle, or they were raised poorly. This is due to the nature of mental diseases, which we understand poorly, and often we associate our mental state as our conscious of our own doing free will. Like someone who is naturally grumpy, we say that's who he is, no one questions that it might be he's got a disease, maybe there is a brain anomaly, or a weird communication of his neurones happening that don't balance out his emotions properly and skew him to being grumpy. But being a grumpy guy would never affect your life enough that you'd seek medical help. So people who are depressed, they're like grumpy guy, except it affects their life so much that we put some research into addressing it. Sad truth is we found very little yet.
With that said, we can see how socially we should be careful how we speak of this, because we're often giving the impression that external factors is the known cause, and that a few lifestyle change is the cure. Except that's false. The cause is unknown, and exercise is just the best we got to help. Imagine if we thought similar of cancer, we'd cut funds for cancer research, we'd just tell people they should eat better, stop smoking and exercise. And everyone who gets cancer got it because they probably didn't follow the right lifestyle.
If I buy a lotto ticket, I have a chance to avoid losing. If I win, it's over. If I lose, it's over. It's only good for one drawing.
However, I have some risk of cancer as long as I am alive. I may be able to do things to reduce that risk, but my chances are infinite as long as I breath. I can choose not to play the lotto, but I can't choose not to get cancer.
The chance you win the jackpot with Lotto is miniscule. It is basically so small, that it isn't worth the investment. You're better off not taking that chance because the risk isn't worth the reward. These games are, basically, fallacies designed to manipulate humans, tricking them into behaviour they should not [logically] resort to.
With cancer, you have genetic risks to get (certain types of) cancer, sure.
However, for example living next to a highway increases the chance you get cancer. So, not living there increases your chance of not getting cancer [at a young(er) stage of your life]. Which increases the goal of human life as we agreed upon.
So I agree they're not the same, but your example of Lotto (or similar games) I find rather bad.
I think with regards to cancer this is related to risk/reward. People take risks such as speeding, eating overburned food, or living near a highway. They feel the risk is worth the reward, and they live under the assumption they get away with it.
This statement is true only in the narrowest sense and is actively harmful to people with mental health issues. Let's compare depression and cancer to see why.
It is common to say something like "cancer is independent of external factors" to mean that anyone can get cancer. It doesn't matter how healthy of a lifestyle you live. Due to random chance, the guy who regularly works out and eats fresh vegetables and avoids toxic substances can get cancer when the guy who watches tv all day, eats nothing but twinkies, and smokes like a chimney might never get cancer. The first guy's cancer was out of his control.
When people talk about depression being "independent of external factors" what they're trying to say is depression can happen to anyone. It doesn't matter how healthy your "mental lifestyle" is. Depression, like cancer, is a disease. Some things may make depression more likely, but anyone can get it at anytime and there's nothing you can do to completely prevent it.
No one feels ashamed for getting treatment for cancer because society recognizes that cancer is disease that can happen to anyone. But people feel ashamed for getting treatment for depression because most people believe "depression doesn't happen to normal people." If you make statements like the parent's about depression but don't make similar statements about cancer, then you are actively contributing to the problem even if your statements are factually correct.