The browser certainly hasn't eaten all of the host OS yet and still depends on it for many low level things but at the same time browsers go far beyond what the OS provides and reimplement many things that 20 years ago would have had you laughed at for implying are part of the browser and not the OS.
Quic still relies on UDP, just like many other non-TCP protocols that are built by vendors for all sorts of applications.
QUIC uses UDP but not in a "it's relying on UDP" way as much as "layering UDP allowed it to go through middleware boxes" way. The developers of QUIC would have liked QUIC to not depend on UDP, and the protocol works fine that way, but then it wouldn't have went through home routers or corporate firewalls very well. Regardless a lot which used to be done in the OS TCP stack is now handled by the browser application. This is similar to how some other apps have used UDP but that doesn't take away from that fact.
Well QUIC is replacing TCP for many HTTP connections now, and that’s implemented entirely in the browser. The OS provides the datagram layer but at this point it’s semantics. The bulk of the protocol is in the browser.
The question was, has a browser has taken anything over from the OS, and the answer is - in this case - yes. It still uses the OS to control access to the network, but the role of the OS in HTTP has been significantly reduced.
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I understand from the other comment in this thread that Google is OK with building an OS within the browser and minimizing the role of the OS, but I do not think Quic is a great example unless and until you consider, as the other commenter did, that it could work without UDP and they used UDP only because the routers and other network hardware required it. So, I understand overall, just knit picking on this particular form of the argument.