Good news! We've improved the UI and APIs for user management within organizations.
If you're using the organization members API to manage the people in your organization, you can now inspect/see the emails of users with the above changes. Either access the service using an Owner/Manager API token or make emails visible within the organization via the "Global Privileges" setting above. When we add Services Bots and Configurable Roles later, we'll allow this to be more granular/specific.
The changes are immediately available for everyone—documentation update to follow.
Let us know if you've got any feedback!