While there's nothing wrong with an analytical or calculating character, if you want more emotional involvement, you'll need to create investment.
1st, you'll need to understand what the player likes about the character & about your world. The 8 kinds of fun, referenced up thread, can help you understand what you're seeing, but your own observations of what engages this player are better.
2nd, you'll need to reward involvement. Reticence is often due to fear of failure, fear of ridicule, vulnerability. So you will need to reassure, comfort, accept and build on the player's offerings and actions. You may need to solicit and reward. "Brian, how does [genasi wizard] feel about what's going on here?" "What does [genasi wizard] do about this monstrous injustice?" Prompt, express reassurance, and build.
3rd, you'll need to nudge or gently push the player to offer this kind of input without overt solicitation. Once they're comfortable, that is. When the other players have done their thing, take a direct moment and ask "Ok, X & Y are doing A & B. How about [genasi wizard]?" This doesn't work before the player becomes comfortable or understands their offers are accepted. It has to be primed and cultivated.
Eventually, you'll have your player trained to respond to prompts with actions of their own. Those actions will show you what the character/player finds important. That's the investment. Play with that a little bit. Try one thing, then another and see what works best. A & B. Is the character just or merciful? Just. Ok, is the player Just or Vengeful? Still Just. Ok, now confront them with injustice. Ask them about the injustice of whatever situation. Impede justice with an obstacle and provoke them to anger at the obstacle.
It comes down to this: find what they want, and put something between them and what they want. That tension will give you the emotional investment you're looking for.
Always, always do so in a reassuring, friendly, and supportive manner. We're not toying with people, we're creating an emotional space where pretending to be a genasi wizard is good silly fun, and not mocked or ridiculed.
-Brad