When I generate a new WinRT project in Visual Studio 2022 I get Error E1696 cannot open source file "winrt/Windows.Foundation.h" yet when I look at the Include directories the files do exist at the correct location.
This is an artifact of the way C++/WinRT works. While the header files do exist in the Windows SDK, that's not where the project goes looking for them. Instead, they are generated on the fly into the source tree under the Generated Files directory.
So to fix the issue you will have to compile a newly created project at least once. This by itself isn't sufficient for IntelliSense to pick up the changes in environment. To help IntelliSense out you're going to have to right-click into the source editor, and select Rescan -> Rescan File.
Once that is done, all the IntelliSense errors go away, including E1696.
Historic background
It's easy to get confused why the C++/WinRT header files are part of the Windows SDK, yet the C++/WinRT VSIX templates aren't using them. A look back at C++/WinRT's history helps explain how we landed in this situation:
Initially, the code generator responsible for producing the language projection header files (modern.exe, later renamed to cppwinrt.exe) wasn't published. Instead, the header files were generated by Kenny Kerr, and published through his modern repo.
Publishing the language projection header files through a GitHub repository carried over into the cppwinrt repo owned by Microsoft, and continued to be the deployment channel for several versions of Windows.
This wasn't exactly convenient for developers, so with the release of the Windows SDK for Windows 10, version 1803 (10.0.17134.0) the headers got added to the SDK (including the code generator). This worked, but wasn't an ideal situation either, as the release cycle of C++/WinRT was now tied to that of the Windows SDK, roughly 6 months.
Decoupling the release cycles was crucial in allowing C++/WinRT to progress at its own pace, shipping frequent updates with improvements and bug fixes. This was enabled by deploying the code generator as part of a NuGet package that the C++/WinRT project templates reference. The MSBuild project drives generation of the language projection headers, and clients can freely decide, which version of the C++/WinRT library they wish to use, controlled through the NuGet package reference.
This is how things work today, but the language projection headers can no longer be yanked from the Windows SDK. They were published, with clients relying on finding them there, and expecting an SDK update to not break their builds. And this is why the Windows SDK contains header files you aren't meant to be using.