They controlled for exercise and the effect size decreased by 40 percent but was still significant. My biggest criticism is, as usual in epidemiology, there's a bunch of confounders that weren't controlled for and are very difficult to control for. Vitamin D levels, social life, psychiatric disorders for example.

However I'm sceptical that vitamin D deficiency explains the entire effect given it is quite large. If sunlight itself isn't relevant I expect other confounders to be stacking on top of vitamin D deficiency.