You probably should have seen something in the status messages for that advertisement. Did you look in the advertisement status node?
Yes. That's where I see that it started the installation but never finished the installation of PACKAGE A.
Beyond that, did you look in the log files on that client?
Mostly irrelevant. I know the installation of PACKAGE A failed. I don't care why at this point, all I care about is that I want SCCM to forget about it and go on with installing PACKAGE B.
How did you verify this? Did you actually look on the client, or did you just confirm the list of advertisements for the relevant collection?
Right click on collection, press PROPERTIES, press ADVERTISEMENTS tab. Before I deleted the advertisement, I could see it in the tab. After deleting the advertisement, it was gone from that tab.
Wait for a policy refresh.
Induced multiple policy refreshes by right-clicking on the collection, pressing SCCM CONFIGMGR TOOLS, and pressing REFRESH MACHINE POLICIES. Each time it gave me the same 10037 error.
Anyway, the reboot worked, and any desire to solve the problem with more finesse than that is merely an academic curiosity at this point.