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.