I don't think anybody is "solving depression" by tackling problems some way or another.
Rather: you're getting out of your depression and that allows you to tackle problems.
Rather: you're getting out of your depression and that allows you to tackle problems.
That makes a clumsy title -- "I solved the particular sub-type of depression I was experiencing with analytical thinking". We don't need that cumbersome language if we just remember that the word depression covers different stuff.
I believe that, yes, you can greatly improve your situation by reducing external stressors. But you'll have to get to a point that allows you to do that first, and when you're basically either sleeping or staring at the wall all day, you are not in that state.
To me, "chicken or egg" is rather clear: the depression must give way enough so you can get things done that will improve it further. Suggesting otherwise sounds dangerously close to "hey, if you just wanted to, you could just get rid of depression" which puts suffering from depression close to "it's a choice" territory.
I feel like OP just used described how he fixed a problem that had him down.
Depression can occur without any real problems or issues. It's not something you can "just fix" by paying your bills.
Adverse life events can trigger depression episodes in some people, and when you’re in an episode, adverse circumstances can both feel worse and lengthen the time in the episode.
Finding a way to solve the problems or avoid them entirely is thus an important part of being as healthy as possible, but it is not in and of itself sufficient for maintaining or returning to health.