No, RN isn't bundled with any browser runtime. You're possibly thinking of Apache Cordova (which leverages the device's browser runtime for rendering UI).
RN is Javascript so it will execute on the device's JS runtime, which I guess is connected to the browser in some way, but RN doesn't use the browser for UI. The rendering is true native UI.
The way React does this is by decoupling the core library from UI rendering. The core React library contains no UI rendering code, it just handles UI lifecycles, internal state, events, etc. for logical UI components as object instances. You then plug that into a renderer library (most commonly the "ReactDOM" lib for web browser rendering).
It's not native, it emulates native UI. What happens when they can't keep up with the updates for all the platforms they plan to support.
Also, I don't think they support platforms other than iOS/Android, desktop and web is planned, but that doesn't help me now.
I do agree with your premise though, but I use React Native for a true native UI, plus it can target Windows, MacOS, Android, iOS, and Web.