Claims for meditation are old and specific. The evidence is recent and narrow, and the two are not addressed to the same question.
The traditions generally do not present meditation as a health intervention at all. Its stated purpose is liberation, union, or the vision of God, and improved wellbeing is treated as incidental or as a distraction to be got past. That matters, because nearly all of the research measures the incidental part.
The research on standardised eight-week mindfulness programmes is now extensive. Meta-analyses generally find moderate effects on anxiety, depression and chronic pain, broadly comparable to other active psychological treatments and considerably smaller than early enthusiasm implied. Methodological criticism is substantial and comes largely from inside the field: small samples, weak or absent active controls, and publication bias have all been documented by meditation researchers themselves.
Adverse effects are documented rather than anecdotal. A minority of people on intensive retreats report severe anxiety, depersonalisation, disrupted sleep, and in rare cases episodes requiring clinical care. Buddhist tradition has always known this, has names for difficult stages of practice, and assumes an experienced teacher present to recognise and manage them. Secular programmes delivered by application to a phone have no such structure, and this is the sharpest practical difference between the traditional and the modern delivery.
The honest summary is that meditation does something measurable and moderate for the average participant in a clinical setting, that it can go badly for a minority, and that neither finding bears much on whether the traditional claims are true, because those claims concern something the studies do not attempt to measure.