I have identified a solution that address this specific issue, as this exact problem affected my environment starting with the June Office Updates.
It appears that Microsoft has changed the communication method for Office Click-to-Run. Previously, OfficeC2Rclient.exe used the WinInet proxy settings (Internet Settings) to connect to Microsoft's cloud services. However, following the June update, OfficeC2Rclient.exe now utilizes the WinHTTP proxy (configured via NetSH) for communication.
Depending on your environment's configuration, systems without a properly configured WinHTTP proxy will default to using the Default Gateway to access the internet. This could result in traffic being blocked by your Default Gateway Firewall, as such traffic typically routes through a proxy. In environments where a WinHTTP proxy is configured to use an actual proxy server, authenticated proxy rules may cause the traffic to fail with a 407 error, as the Windows WinHTTP proxy does not handle authentication well.
In our environment, we addressed this by reconfiguring our firewalls and proxies to allow the necessary communication. Given the risks associated with recent CVEs affecting Office products in the past couple months, I hope this information helps others who have been unable to update their Office installations, thereby reducing their exposure to potential vulnerabilities.