macOS Big Sur Installation Failed: "An Error Occurred While Installing the Selected Updates"

Apple this morning released macOS Big Sur to the public, and since the software went live, it has had extremely long download times that have prevented many from being able to install it.
Even those who have made progress through the download have run into problems where the software downloading process is suddenly interrupted and must be restarted. Others have reported being able to download the update and then having the installation fail. Apple's site appears to be having some major issues du