I wouldn't recommend LSD to people with anxiety, as raises anxiety. I would _mildly_ recommend it to people who suffer with depression, but have a low level of anxiety. And yes: it does help break out of rumination cycles, and weaken old neural connections.
You make it sound like negative thoughts are the _cause_ of depression. This is not the case, although they are correlated: when you're depressed, you tend to have negative thoughts, which can drive you deeper into that very depression.
What I would recommend strongly to people with depression is a prescription anti-depressant (it can take a long time working with a good psychiatrist to find one). Possibly light smoking as well, if the person is otherwise healthy.
There is no silver bullet here. Certain people have certain imbalances in neural chemistry, and small doses of the right prescription drug can help correct that. For chronic symptoms, people have to be on the drug for a lifetime; for others, a few years.