Why do you focus heavy on Lucifer's depression?
Because it's a key core element to his character. You can't just have him as the silly, comic relief guy who's Sad™ and tries to be a Good Dad™.
Depression bears its ugly teeth like glass fragments on the floor from a shattered mirror. It's easier to tell yourself you're not than to accept the fact that you are. Lucifer is a hypocrite. He's far from innocent and kind. He's fully embodied the devil side to himself, long forgetting what he once was as an angel. You get too saturated into the part you play and gaslight yourself into thinking it's the reality. He believes it. This is who he is and always has been.
But isn't that how a devil should be? Handsome, a gentleman, a facade of kindness? Lures you in to EAT YOU UP?
The lines begin to blur of who you are and who you once were; depression is a heavy burden to bear if you have to carry the load by yourself with a non-existent support system. And maybe through the Sins, he could've had one. But depression speaks of cruelty in the ears of the believer. The people who you love and care about: don't care about you. So why should you care too?
Does this mean it's factual? No. But it isolates you; creating a false narrative that you start to believe. Lucifer hates himself, hates the person he's become and he doesn't know what to do about it or how to change it. He was never shown, never taught. Only expected. So he submits, copes, wears the very mask he resents sinners wearing.
And though his hate for himself rings strong, he'll always see himself better and above those who actually deserve to be in Hell.
Once he opens up, once he allows people into the cage he's locked himself inside and allow his facade to shatter, will he finally take the steps in the right direction to finding himself again.