After downloading from the App Store, the installer opens automatically. Safari uses these links to find the old installers in the App Store. It takes time to download and install macOS, so make sure that you're plugged into AC power and have a reliable internet connection.
Time Machine makes it simple, and other backup methods are also available. If a macOS installer can't be used on your Mac, the installer will let you know. Apple recommends that you always use the latest macOS that is compatible with your Mac. To get the latest features and maintain the security, stability, compatibility, and performance of your Mac, it's important to keep your software up to date.