Old hardcore quake3 player here. Q3 didn't wait for a server response, but the net-code was not that great and the interpolation did iirc not really take into account server responses. You could end up with players teleporting because of network issues. OSP was the first mod iirc that tried to fix it and it was somewhat successful.

The one mod that made great strides to fix the issues with the Q3 net-code was CPM (www.promode.org). 50-70ms to the server felt like 20-30ms in Vanilla Q3 and when you hit 20-30ms in CPM it felt like LAN play in VQ3. It fixed the niggling issues of lost packages (caused players to warp) as well. You even had some players that intentionally downloaded things in the background to cause dropped/delayed packets so that they would warp.

Eventually the net-code in CPM got so good the community even had cross-Atlantic competitions. The team I played in had 100-120 ping vs west-coast American teams on NYC servers and it was actually playable to the point where you could have fairly fair fights with them. Except against Team Abuse... :p. Man what a schooling in team-play and lock-downs of maps they gave my team.