Earlier quoted context omitted.
The bit where Apple's OS tries to connect to Apple's update servers, and can't authenticate because Apple switched to an incompatible root CA.
Ah yeah, I've recently received for free an iMac running Macos 10.9. It's simply impossible to upgrade; the only proposed upgrade release is 10.11, the installation starts then fails in a loop. Fortunately I don't actually need to save anything from this machine, and I have another Mac to download a newer OS installer, but that's quite painful.
If the default/upgrade installation is failing then I'd try creating a bootable installer on USB [1]. If it still fails then try erasing the target drive first to do a clean install (you can do this by running Disk Utility from within the installer).
[1] Instructions here: https://support.apple.com/HT201372