The system API problem can be solved by shipping automatically generated API bindings (similar to what Microsoft does with the Rust bindings to the Windows APIs, a similar project for Zig is underway). Once those bindings exist, no Windows SDK is needed anymore and cross-compiling should "just work".
PS: Zig actually ships with the mingw headers in the default installation (e.g. windows.h, d3d11.h, dxgi.h etc are all there), I haven't actually tried to cross-compile a Windows executable using the Win32 APIs though, but it looks like it should work already.
Reverse engineering is legal in the US (unless bound by contract).
And to go back to my examples, headers and libraries alone won't make it, without the remaining toolchain to create the appropriate package.
If you want to cross compile C stuff e.g. a Win32 GUI app with Direct X or get OpenGL bindings for SDL though those all work just fine if you provide the SDK files same as you would compiling natively. The advantage being now you can do all builds from a single build instance just by feeding it the libraries and a target platform.
I don't see shipping all targets all the time to be a particularly scalable solution in general. When you start including all libraries necessary to develop actual apps, the download size will eventually get so big that people will complain. Just make it easy for developers to get the targets they need, but don't ship everything by default.
For Mach (the Zig game engine I'm working on), I have a Zig build script which fetches the required parts of the macOS and iOS SDKs from GitHub to enable cross compilation from Windows/Linux -> macOS. Zig's linker, zld, enables that to work easily.[0]
For SDL.zig, they have also achieved cross compilation for most targets out of the box[1].
Contrast this to say, Go or Rust, where most GLFW and SDL bindings require mucking around with the system dependencies. It doesn't just work out of the box, I think that's really unfortunate and hope other languages work more towards painless installation of C-binding dependencies.
It seems to be what Flutter is doing, and it must be a ton of work.
E.g. try to cross compile Metal shading example for iOS from Linux with zig, or do a Win2D UWP rendering application from Linux with zig.