The reason Calibre does not have an official Appimage or Flatpak is because Calibre embeds itself all its dependencies rather than using a dependency tree, aka Appimage/Flatpak install the say dependcies individualy so that can be updated outside of Calibre.
The advantage of the official method is that it allows Calibre to be much more stable and easier to install on a wide variety of distro. Unfortunately, however, this approach is not very Linux-like, and therefore it not allowed.
The main conter-part is that the official script need to be manualy run at every update, not that much, but it a thing.
Depending of your distro, you can get unoficial package that support auto-update and use the offical script:
Appimage:
ivan-hc/Calibre-appimage
Arch Aur:
calibre-bin
This two are updated in 24h range.