docs(auth): update examples to use modular API

[DoctorJohn]

Description

This PR updates the authentication docs to use the modular API.

I just finished migrating an app to the modular API and put my insights into this PR hoping the next person can benefit from it.

Checklist

• I read the Contributor Guide and followed the process outlined there for submitting PRs.
  • Yes
• My change supports the following platforms;
  • Android
  • iOS
• My change includes tests;
  • e2e tests added or updated in packages/\*\*/e2e
  • jest tests added or updated in packages/\*\*/__tests__
• I have updated TypeScript types that are affected by my change.
• This is a breaking change;
  • Yes
  • No

Test Plan

I'll wait for the Vercel preview and then read the docs again. I already cross-checked the edits I made with my own code after the migration.

🔥

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Name	Status	Preview	Comments	Updated (UTC)
react-native-firebase	✅ Ready (Inspect)	Visit Preview	💬 Add feedback	Feb 20, 2025 6:14pm