Accessibility: Site Health: Improve the "Copy site info" button accessibility.

- avoids a focus loss when clicking the "Copy site info" button
- uses `setTimeout()` and `clearTimeout()` to properly handle the "Copied!" text
- minor JavaScript coding standards

Props audrasjb, Clorith, afercia.
See #48463, #50335.
Fixes #50322.

Built from https://develop.svn.wordpress.org/trunk@48233


git-svn-id: https://core.svn.wordpress.org/trunk@48002 1a063a9b-81f0-0310-95a4-ce76da25c4cd