After looking at the configuration in the debugger I found the answer.
Serilog__WriteTo__0__Args__path
(All platforms)Serilog:WriteTo:0:Args:path
(Windows)Serilog--WriteTo--0--Args--path
(sourced From Azure Key Vault)
Note: The Configuration in ASP.NET Core documentation now covers this.
So I need to use the array index (zero-based) as if it were a name.
Here is the screenshot of the debugger, also (thanks to Victor Hurdugaci in the comments), the unit tests are a good place to look for examples.