[native] Move mode/nextMode state in LoggedOutModal to function component

Summary:
As a first step towards replacing the Reanimated 1 syntax in LoggedOutModal with Reanimated 2 syntax (ENG-8146), I'm converting the class component to a function component.

Depends on D12180

Test Plan:
I tested this diff stack by playing around with LoggedOutModal on my iOS simulator in my local environment:

1. I navigated to LogInPanel and back
2. I navigated to LegacyRegisterPanel and back
3. I navigated to FullscreenSIWEPanel and back
4. I navigated to QRCodeSignInNavigator and back
5. I navigated to Registration and back

Reviewers: inka

Reviewed By: inka

Subscribers: tomek

Differential Revision: https://phab.comm.dev/D12181