> can't the exception identify what the runtime type of T is for the particular call
No, it does not identify it. Even Exception.TargetSite seems to reference the "generic method definition" (with the T) instead of the constructed generic method.
Note that these stack traces do not include the values for ordinary arguments either, so this is not terribly different.
On the other hand, the Visual Studio debugger will display argument values for the current call stack -- include the types bound to generic arguments like T. This is an unmanaged debugger capability and AFAIK not something that you can do from within your code.