They can trigger a paradigm shift in many other areas, not just mental health care. Except it's so hard to explain this to a 'normal' person, who's never taken any. "If God were to permit you a brief voyage into the divine process, let you whirl for a second into the atomic nucleus or spin you out on a light-year trip through the galaxies, how on earth would you describe what you saw when you got back, breathless, to…
But in real life, seeing as how one's predispositions get highlighted with acid, we'd probably end up with a lot of people thinking they now know the answers. Or that their trip showed them that (Allah|Zeus|Quantum Democracy|Whatever) is the True God and go all nuts on that.
Leary and others made up these unsubstantiated claims about the effects of acid and really tried to sell it as something beyond what it is. It's a damn awesome molecule that does really nifty shit to brains, and certainly bears plenty of research and use.
But to say it can cure cancer? Come the fuck on man. That's just new-age heebie jeebie nonsense with no basis in reality. It's why a lot of pro-LSD people are simply not taken seriously, because they forget it's just a molecule messing with brains' perceptions.
(Unless you meant cure cancer in the "well it made you feel better, and feeling good is important to health so it might give you a better chance against some diseases", in which case it's not much of a claim and certainly shouldn't be phrased that way. Hey, Apple products can prevent street crime, because the metal bodies might slow down bullets and make an otherwise fatal shot survivable!)