He was reticent and more humorless than normal, but that's not the same thing as being depressed.
It's a fan interpretation similar to "Emo Dante" for the TGS10 Dante. Unlike that description, Dante being depressed is a way to explain the changes to him in 2 and the anime. It's not meant as an insult but comes off as headcanon.
Capcom never figured out what to do with Dante after 1. They found their footing with Dante as an extroverted joker than an introvert. So I rather keep him as one but make him less cringy and give more variety.
So far both DMC2 and the anime's take on Dante feel empty. You can keep him as an introvert but you'd have to give him a real inner life for that to work.
There was a rumor that Dante wasn't the original protagonist for DMC2. Kariya wanted to swap protagonist and there was a rumor of a guy with a green jacket. So I don't mind replacing Dante for Trish, Vergil or Nero. It's not essential but a more interesting opportunity.
The series struggles with the case of the week setup. Dante outside of his family trauma tends to feel like a character you would cut. So you could cut Dante or use him better. Both are valid and I'm hoping the upcoming animated series does the latter.
DMC2 was cobbled together from a bunch of pitches lying around. So it ends feeling less put together than the other games. So remaking it FF7R style is a bigger task.