Nothing stops them, but it would be difficult for them to get away with it without being found out. Many paranoid users run monitoring software (e.g. Little Snitch for Macs, OpenSnitch for Linux, similar tools for Windows) that would detect the browser opening extra network connections to send this information back to the control site. If the browser vendor is in league with the OS vendor (or they're the same, e.g. Microsoft provides both Windows and Edge, Apple provides both MacOS and Safari), they may be able to add additional measures to the OS to thwart this monitoring. But this suffers from some flaws:
- It doesn't work for users who run the browser on another OS.
- Users can run monitoring software on their router, out of reach of the OS.
Even if you posit a conspiracy between the OS/browser provider and router manufacturers, there are third-party, open-source router firmwares.
And if the browser is open source, as in the case of the Chromium core of Google Chrome and Edge, and Mozilla Firefox, it would be found in the public source code. Although in the cases of Chrome and Edge, it might be outside the open-source parts.
As long as there's healthy competition in the browser market, a provider who does this would just be shooting themselves in the foot. When they're caught, the users would just switch to a different, safer browser. There's almost never been a time when users didn't have a choice of several browsers, except maybe the first year or so of the web's existence.