Authors of studies like this are well aware of this. I haven't read this paper, but these papers often go to great pains to explain how they controlled or screened-out other causes.
I think it's fair to be off-hand sceptical without digging into the details of the paper. Since we don't really understand depression, it seems fairly obvious that it would be nigh on impossible to control for causes we don't know.
While “correlation is not causation” is a tired refrain, I agree that default skepticism is the generally appropriate response. There are peer reviewed scientific studies showing essentially everything and anything you could want to believe. Relatively few hold up to scrutiny or repeat experiments.
There is undoubtedly an opposing study that shows increased screen time correlates to happiness.