The Big Sur installer was (is?) a mess for some people & systems. It took me two days to get Big Sur installed to a MBPro 2016 laptop after fighting with cryptic error messages. For anyone having issues running the Big Sur installer from within macOS you should try creating a macOS Big Sur USB installer which should bypass those Gatekeeper error messages:
https://support.apple.com/en-us/HT201372
Or you can try booting into Internet Recovery Mode using Command + Option + R to access the online Big Sur installer:
https://support.apple.com/kb/HT204904
I highly recommend using the bootable macOS USB installer method as it eliminates (or at least minimizes) networking issues.
I also highly recommend you have a good backup before attempting to upgrade to Big Sur in case something goes wrong during the upgrade process or in case you decide you have apps or hardware that is not compatible or you just don't like Big Sur.
Edit: Also make sure to download the most recent version of the Big Sur installer since maybe the latest 11.1 version is better. I used the v11.0.1 installer when I had the issues. Check your Applications folder for the "Install macOS Big Sur" installer and delete it and empty the Trash so you can download the latest version of the installer.