Solution: Stumbled upon this article by Ed Williams. It appears Microsoft has a hotfix for this, but it seemed easier to apply the suggested registry fix. This creates a service dependency on the HTTP service so it waits for the CRYPTOSVC to start. This has resolved all our SharePoint hanging issues.
To work around the issue without installing the hotfix, create a DependOnService registry key to modify the behavior of HTTP.sys. This makes HTTP.sys depend on crytosvc service to be started first. To do this, follow these steps:
1. Click StartStart button, type regedit in the Start Search box, and then press ENTER. If you are prompted for an administrator password or for confirmation, type the password, or provide confirmation.
1. Click StartStart button, type regedit in the Start Search box, and then press ENTER. If you are prompted for an administrator password or for confirmation, type the password, or provide confirmation.
2. Locate and then click the following registry subkey:HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Services\HTTP
3. On the Edit menu, point to New, and then click Multi-string Value.
4. Type DependOnService, and then press ENTER.
5. Right-click DependOnService, and then click Modify.
6. In the Value data box, type CRYPTSVC, and then click OK.
7. Exit Registry Editor.
8. Restart the computer.
Thanks to Ed and to Colorado Springs SharePoint!