I tried your project on a current fedora 38 with dotnet 6 / 7 installed. Unfortunately it did not work out. It looks like the code in GtkSharp is hard coding an old webkitgtk library.
Unhandled exception. System.TypeInitializationException: The type initializer for 'WebKit.WebView' threw an exception.
---> System.DllNotFoundException: Webkit: libwebkit2gtk-4.0.dll, libwebkit2gtk-4.0.so.37, libwebkit2gtk-4.0.dylib, libwebkit2gtk-4.0.0.dll
at GLibrary.Load(Library library)
at WebKit.WebView..cctor()
As this project looks quite interesting, I wonder what your thoughts are in regard to base it on Gir.Core? Gir.Core supports Gtk4 and current WebKitGtk versions. It follows the binaries inside the current gnome Flatpak SDK. This allows to create Flatpaks which can be distributed via Flathub.
But the project and API is not complete or guaranteed to be stable, so it is quite possible that the APIs you are relying on are not yet generated or are even not supported anymore as there was some API break on WebKitGTK side.
As I'm actively working on Gir.Core, I would be interested in making the needed API available to you.