The answer to this is complicated.
Both Google Chrome and Microsoft Edge actually implement this. Behind the scenes, both will upload your browser history to the cloud. You can see it in network packet captures. It's implemented in the browser for the vendor, but not for the user.
The choice to not implement this for the user is very deliberate. It's contrary to the vendor's interests if the browser provides this capability directly to users. If a user's browser can take you to a website directly, then you are not using the vendor's search engine, meaning you are not looking at their ads, paid search results, algorithm, etc. It would severly impact their business model.
This is also the reason why browsers have:
- Adopted Google Chrome's "Omnibar" instead of a separate address bar and search bar.
- Implement only basic hierarchical organization for browser Favorites.
Directly and indirectly, Google is the central nexus of all modern browsers. Aside from Google Chrome, they also:
- Fund the vast majority of Firefox.
- Pay Apple for preferential treatment.
- Provide the same mechanisms to vendors who base their browsers on Chromium (i.e., Microsoft Edge, Brave).
I would love for this to not be the case. There is hope to be found in small independent browser and search companies/projects.