Maybe "significantly" is too positive a word, ultimately that's a question of semantics. Either way, the other benefits are very likely to make the surgery worth it for people. The person I know has told me how they felt literally trapped inside their own body before as the obesity made moving around hard and painful, triggering their PTSD which probably shares an origin with the obesity. In my experience (and your mileage may vary) it has made them a much happier person.

I'm not saying that everyone who are unhappy with their weight should have bariatric surgery. It is not an easy and risk-free solution and an amount of gatekeeping is warranted. However it does have its place as a treatment for patients who are impaired by their obesity an for whom less radical methods has proven ineffective.