This is what has happened so far. I tried Windows update direct from Microsoft, and kept the security roll-ups separate. I successfully installed 8 other packages first, then tried the June roll-up KB4503292. It failed with the same error code. I went back to the local WSUS, which again wanted to install the November (KB4467107) and May (KB4499164) security roll-ups, but failed again. This is my work computer and I ran into a weekend.
I went back to step one, and began following the steps given in the "follow these steps before posting". I was getting inconsistent results and I wanted to get a consistent, replicable story. I found that every time I ran "scf /scannow" it would return either "found corrupt files and was able to repair them" or "found corrupt files and was unable to repair them". It would flip between the two outcomes each time I ran it. I re-ran KB947821 (System Update Readiness Tool). It gave a completely clean report with no errors. After this, I re-ran scffix. Again a completely clean report. Then I re-tried KB4467107 and KB4499164. It still failed with error 80004005.
The PC is a desktop device from HP and I have the DVD's for re-installing the operating system. But the only option is a clean re-install after wiping the hard drive. I am thinking I might have to do an inventory of all the software I have installed and make sure I can re-install every one.