If you truly want the most feature-rich experience of a browser, then you probably need to use the one on your computer. This is because even on mobile, certain features available on desktop aren’t supported, like extensions, for example. This gets progressively worse on other devices such as consoles, in which browsers are there for the basic stuff.
However, if you were hoping for a more full-fledged browsing experience, a report from The Verge has revealed that Microsoft has started to publicly test its Edge browser on its Xbox consoles. Note that this is the Chromium-based version of Edge, not the original that shipped with Windows 10, which means that it should be more capable than the default one on your Xbox console.
Given that it is built on Chromium, it means that in addition to playing Xbox games on your console, you will also be able to stream games through the browser from other platforms like Google Stadia. It should also help make web-based services even better, like Discord or Skype if you use those for communication.
At the moment, the test is limited to Xbox testers in the Alpha Skip-Ahead group which means that not everyone will be able to take it for a spin, but hopefully it won’t take Microsoft too long to release it to the public.