Generally, no. You can't become better by taking a drug, but they're not talking about bettering themselves. They're talking about healing. If your intention is to better yourself with the aid of psychedelic drugs[1], you'll only get out what you put in, and psychedelics will mostly only inspire you to put forth the effort. edit: You still have to work at improving yourself through other means.
> For each individual who says psychedelics have made them a better person, there are hundreds who make the same claim about religion.
Well, not that many people are willing to try psychedelics.
> And I think we can all agree that the risk of biological side effects is much lower with religion.
Every year more people are killed in the name of religion than by psychedelics. Psychedelics kill noone. Their LD50 is usually too high.
> If I were going to follow your advice and try MDMA, shouldn't I follow my father's advice and try religion first?
Sure, but please try more than one religion, and check out philosophy too. You can choose whether or not to try MDMA in the same way that you can choose whether or not to go sky diving.
[1] This is not an inherently misguided idea, but in uncontrolled environments, psychedelics can be quite dangerous.