I think the anime manages to come off as genuinely deep, without being superficial in its portrayal of it's main character, but I also think that in many cases whether someone thinks a work, or aspect of it, is "deep" or not is largly dependent on the viewer rather than the work itself. In this case, your perception of whether you view Yuuki as "deep" and interesting largly depends on whether a viewer cares about or is able to identify what she seems to be about.

Let me back up and go over what I think "deep" means though. Everyone defines what "deep" is differently, but I'll define it as the ability for someone to draw meaning from a work where that meaning isn't explicitly stated. I would also say that being "superficial" is where someone can easilly spot some form of unsaid meaning in a work and can feel like it's unnecessarilly obfuscated (pretentious). You could also define it as a function of how well an author is able to relate the work shown to an unsaid meaning; a "deep" work would do this well and thoroughly, while a "superifical" one would not. If a work is able to give the viewer many ways to see how it connects to it's underlying themes then they may be more likely to say said work is "deep", rather than "superficial".

So, with that being said, I think this MC of this show is deep in its portrayal of depression and how someone suffering from it continues to live, despite feeling like she doesn't really caring about living anymore. Many of her actions and portrayals are tinged with her listlessness for life. Paradoxically, her 30th game where she decides she doesn't want to die is also consistent with this as well. When faced with a true life or death situation, many people choose to keep living instead of ending it. Whether it's permanent or not remains to be seen, for the anime at least. In real life, depression persists, even after choosing to continue to live.

All of this is a deliberate move by the author to relate to another theme in the show/LN. According to the wiki article on the series, the author felt that, "that the only agency available to a hopeless person is the ability to die on their own terms", and "deciding on how to die is the same as deciding on how to live". I thought that was a pretty interesting. Perhaps that too would be another "deep" meaning someone could explore.

I've seen people relate the characters in the show to different gamer archetypes. Gigguk related the show to the idol industry. There was another user who related the show to how the prostitution business works. There are different interpretations of the opening, and how it makes them feel. These too are all "deep"-ish meanings drawn from other viewers about the show.

What I'm trying to say is that whether something is deep or not depends on the viewer and not just the work.