I'll add to this conversation. I have one computer out of 6 that is exhibiting the same symptoms.
I've spent about 5-6 hours of quality time with the problem computer trying to get the update to run.
I've tried the usual methods of running of the Windows Troubleshooter, renaming the SoftwareDistribution folder procedure, and trying an in place install using the downloaded files from the Microsoft site.
The update process fails as previously described -- stops at 100% for a period of time then the message "Some update files are missing or have problems. We'll try to download the update again later. Error code: (0x80073712)" appears
When I try the in place update it appears to run, then, the message indicating that the update was not installed appears.
One thing that I do notice is that the software download folder appears to be cached somewhere in the system. I'm on DSL as an internet connection and download speed is limited to 6 Mb/S so the 600 MB update download will normally take about 10 minutes. HOWEVER, when I rename the SoftwareDistribution folder or just empty the downloads folder, the entire 600 MB update file appears within about 30 seconds of starting the update. That could not be downloaded from the internet as my connection is just not that fast. I turned off the option to download from other computers on the network but the file still appears nearly immediately. So, if anyone has an idea of alternate places Windows could cache the update files I would be interested. That would explain why renaming the folder does not work as the cached update file would be the corrupt one and is not re-downloaded just cached elsewhere and restored to the download folder. However, that's just speculation. It also does not explain why the in place update attempt fails.
Antivirus is ESET, but behavior is same with it enabled or disabled.
(minor update) Windows is 21H1 Version 10.0.19043 Build 19043
Not looking forward to re-installing Windows on this machine so I hope there is a constructive suggestion.
Anyway, that's my report.
UPDATE: 7/20/21
UPDATED 7/21
--> I correct my syntax mistake below. If you're getting error 2 out of DISM it's because I had online and offline syntax in the same line and it just doesn't work. Of course I repeated the error via cut and paste.
Corrections below:
I finally got the update (KB5004237) to install. The key was probably running: DISM /Online /Cleanup-Image /RestoreHealth
However, several other steps were done. YMMV, but they are good to do.
First I reset the Windows Store (wsreset.exe). It's a generally good thing to do as the store has been the source of several strange aggravating problems.
That's probably why there's a reset command.
Second, I was able to cleanup the download cache by first stopping the windows update services (net stop bits, net stop wuauserv, net stop cryptsvc)
That allowed me to run "Disk Cleanup" and remove the cached system update files. Otherwise Disk Cleanup will run but the files are not touched. If you do stop the update services, be sure to restart the services (net start <blah>) before continuing.
After deleting the cache, the update took a normal 15 minutes to download. However, the update again failed as before -- 0x80073712
Continuing:
So, I ran the Dism /Online /Cleanup-Image /RestoreHealth command with the options shown. That command ran and stopped for a lengthy time at 62% but eventually completed.
It prompted that the command completed successfully indicating that it did something. The log is long, I'll probably peruse it sometime, but not now.
Running Dism /Online /Cleanup-Image /ScanHealth and Dism /Online /Cleanup-Image /CheckHealth then indicated no problems.
I then ran SFC /scannow which reported no problems.
Finally, I retried running the update. This time it took longer to run, stopped at 74% for a while, then climbed up to 100% and gave the happy "Restart?" prompt.
So, briefly, I'd delete the download folder files and cache and get a clean start. Re-download the update files. Run DISM as described above, then check with SFC /scannow. Finally retry the update.
Windows being what it is, this may or may not help your particular case, but it's worth a try. The symptoms are those of some kind of file corruption, I have had a power outage recently and that may have corrupted a file. If so, DISM will try and fix that. It's a consistent story anyway.
Good luck!
Quick note: The above will fix an online image. Here's a link to repair an offline image with an online and vice versa in case your image is really borked:
https://docs.microsoft.com/en-us/windows-hardware/manufacture/desktop/repair-a-windows-image