Thanks for putting the effort. Though my knowledge is limited but I don't get this. Why React is different from React Native? Ideally web should be another target just like iOS or Android.
You're correct that there's no theoretical reason that RN apps can't run on the web; as dabbott mentions, react-native-web is one third-party attempt at this. We'd like to support this officially one day but don't currently have a high-quality solution we can recommend.
Ideally web is just another taraget :) https://github.com/necolas/react-native-web
That said, while writing a website with react-native-web is potentially interesting for new sites, it can be less worthwhile for an existing site to just embed a little bit (which is one of the huge values of React). It's also still very experimental. But I suspect it will be supported in a huge way soon.