He also says that Carmack re-wrote the quake network engine which is true, but links to and focuses on Valve's lag-compensation which went far further than what quakeworld did.

The quake re-write was Quakeworld[1], which didn't have valve's lag compensation for shooting. The crucial thing that quakeworld did was de-couple client and server for movement. The result was that when you fired you would see your shot locally, but any effects from that such as blood wouldn't be seen until you got a response from the server. You would still have to lead your shots (shoot ahead of moving enemies) in quakeworld to hit on a modem.

Even Half-life (based on the Quake 1 engine) didn't have valve's lag compensation until the half-life 1.1 patch in 2000, two years after half-life was first released.

[1] https://en.wikipedia.org/wiki/QuakeWorld