[native] Extract the buttons from the logged out modal

Summary:
These buttons will be used later in the stack. Use the primary buttons instead of the old styling.

Before

After

Depends on D14064

Test Plan: Check if the logged out modal still looks correct.

Reviewers: kamil, angelika, ashoat

Reviewed By: ashoat

Subscribers: ashoat

Differential Revision: https://phab.comm.dev/D14066