When I go hiking for a couple of weeks (with significant elevation gain each day), I eat like a pig. I mean, something like 5000 calories a day. I lose weight over that time. So it's possible to lose weight through exercise, however you really do have to spend a very significant chunk of each day doing it for it to work.
Not eating as much takes much less of your busy day.
If you are exercising on a caloric deficit, you are going to feel constantly tired and shitty. I find it way easier to just be on a caloric deficit.