This kind of makes sense. Most forms of depression have to do with not wanting to do something or lacking motivation to do so. Whereas in sports, people get busy usually, they do something, even if it may just be soccer playing. Exercise is also usually good for one's physical well-being, blood circulation increases, muscles may improve, pain may go away (depends on the exercise, but usually say, two days after some heavy work out, most people may feel better than before, unless it was some extreme exercise that caused injury).

I don't think this works for all type of depression though.

I wonder if there’s some survivorship bias. E.g. did they only study people who started and maintained a routine vs. everyone who started a routine but couldn’t stick with it due to depression symptoms

Observation bias. Those who succeed to drive themselves to do an effort, are already out of the depression. Try doing 7hrs sports a week and falling into depression: You won’t have the energy. It’s not sports that gets you out of depression, but sure it’s a stage on the way back and you gotta constantly give it a chance.