The study talks about "positive self-statements"; that is, it asks subjects to repeat positive statements about themselves. This is a very specific action that, apparently, does no good. The NHS and BBC News articles, however, talk about "self help", which is a broad category that includes positive self-statements, but also includes lots of other, different things, which the study says nothing about.

Second, the study measured participants' self-esteem, but did not measure any aspect of their behavior. If positive self-statements lowered self-esteem but increased desired behaviors, then they would be a net benefit. In fact, that is the result I would expect (albeit given the benefit of hindsight).