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This is a feature actually implemented by your web server, not Admin Tools itself. We just create the .htaccess and .htpasswd files for your web server to apply this feature.

If there is a redirection e.g. from HTTP to HTTPS, or from www to non-www (or vice versa) it would invalidate the authentication and ask you to log into your site again. That's how browsers work. They save credentials based on protocol and fully qualified domain name, meaning that http://www.example.com, https://www.example.com, http://example.com, and https://example.com will be different authentication domains for most browsers (at least Firefox and all Chromium-based browsers which accounts for 99% of browsers out there).

For what it's worth, you do not have to reinstall Joomla! to remove that protection. That's like dropping a nuclear bomb to kill an ant. Yes, the ant is gone, but so is half of the country. Instead, just delete the administrator/.htaccess file, as we explain in the documentation.

Finally note that it's possible that your web server is using a configuration which Admin Tools' Administrator Password Protection cannot support. In this case, don't use Admin Tools' feature. Instead, ask your host how you can password-protect the administrator directory of your site. I have yet to see a host which does not have this kind of feature in their hosting control panel; it's one of the basics that's been around since the late 1990s and the dawn of commercial web hosting companies. Since that tool is made to work with the exact configuration of your web server it is guaranteed to work (or will be made to work by your host), whereas Admin Tools can only support most, but not all, server configurations as it can not made to support just a particular hosting setup.