Windows 10 computer won't update

My PC (Windows 10 2004) will happily install all updates provided via the company WSUS.
But (in contrast to all other company client PCs) it fails to instal updates from Microsoft directly.

It keeps saying that "important security and quality fixes are missing" (which is the reason why I wanted to check directly with MS in the first place), but when I click "Search for updates online with Microsoft", it fails with a message that no connection with the update service could be made. I should check my internet connectivity (as if …)

Windows troubleshooting for updates says that it cannot find any problems.

A (tiny) bit more helpful is the accompanying event: Id 36871, source SChannel, with internal error status 10013. Google results for this suggest various things such as

• enable SSLv3
• Make .NET 3.5 use TLS1.2
• clear system certificate list
• check clock synchronization
• check permissions to c:ProgramDataMicrosoftCryptoRSAMachineKeys

None of these helped in my case.
What else can I do? Is there a less generic interpretation of the event?

I also tried to use Wireshark and see what happens during the failed attempts. It seems as if a few https (TLS1.2) connections with slscr.update.microsoft.com are being established successfully, but with merely about 64 bytes of payload. By their nature, I can’t look deeper into the conversations.