Hi @powerout,
Thank you for the console screenshot — that was exactly what we needed, and you have found a real bug.
The Close and Show again in 3 days links only ever worked when the background request that stores your choice came back successfully. If that request is blocked or rejected on a site — and on yours something is blocking requests, which is what the ERR_BLOCKED_BY_CLIENT line in your console shows — the click does nothing at all and reports nothing, so the notice comes straight back on every page load. The links themselves have no plain fallback, so there was no other way to get rid of it. Sorry about that.
What we are doing about it
We are removing that notice entirely rather than repairing it. The message it carries (that you already have another backup plugin installed, and that WP Staging is worth comparing against it) is already part of the newer first-run experience, so the standalone popup is redundant. It will be gone in an upcoming WP Staging release.
How to get rid of it right now
Either of these works on the version you have installed today.
The tidiest option is a small snippet. Add it to your theme’s functions.php, or, better, to a file such as wp-content/mu-plugins/hide-wpstg-notice.php:
<?php
add_filter('wpstg.notice.hide_backup_notice', '__return_true');
That filter is built into the plugin and hides the notice immediately, with no other effect.
If you would rather just mark it as closed and you have WP-CLI available, this does the same thing permanently:
wp option update wpstg_backup_notice_is_closed 1
Once you are on the release that removes the notice you can delete the snippet again — it will simply have nothing left to do.
Thanks again for reporting it, and for taking the time to capture the console output.