There's a semantics problem that's actually relevant with all this: What do you mean by "game designer"? Because that's like saying "I wanna be a scientist" or "I wanna work with computers" in the sense that it's a wide umbrella term.
"Game Designer"
- Systems Design (inventory, combat, etc)
- Level Design (mapping, interaction, puzzles, etc)
- World Design (how things look, why things look, where things exist, etc)
- Character Design (costumes, hair styles, vibe, etc)
- Writer (storyline, dialogue, events etc)
- Artist (physical art assets, pictures, graphics, etc)
- Composer (music, which music when, etc)
- Audio Effects (sound effects, UI noises, etc)
- UI Design (what buttons go where, which layouts are most intuitive, etc)
The list goes on, and you can even break all these up into sub-categories or oversee each part with a director, even going as far as adding in another person as a coordinator between areas of expertise.
There's nothing stopping you from trying to do it all yourself. Though, in reality, even solo devs rely on assets from other independents. This very website is a good case and point. In that regard, you'd be acting as a directer, designer, and self-coordinator when it comes to anything not physically done yourself (like adding in your favorite musician's sweet background music).
Solo devs do it all, and move slow to pay the passion tax. Small teams will have multi-talented individuals. Myst was made with 2 guys, an artist and a guy who did everything else... and while that might sound unbalanced on the surface, I can tell you from experience I wish I had a friend like that lol!