Why can't I find a specific person to designate as my proxy?

That's because the person you want to designate as a proxy doesn't have the Impersonate User (FND_IMPERSONATE_USER_PRIV) privilege. Ask your administrator to assign the privilege to that person.

Note: If you're an HCM Cloud user, remember that your proxies will have access to your personal and sensitive information, such as your salary details, national IDs, and so on.