Upon further experimentation, the Schannel Error Codes don’t appear to correlate with ClientPatcher.exe freezing up, since they only appeared a few minutes later. On the plus side, my Windows 7 is now more secure; I investigated Advanced Internet Properties and switched TLS 1.0 to 1.2.
The Microsoft Event Viewer was a bit of a bust. The Windows Logs for Application, Security and Setup showed no events correlating with the freeze-up. Under System, 2 appeared at that time: “The Application Experience service entered the running state.” and “The WinHTTP Web Proxy Auto-Discovery Service service entered the running state.”