From what I heard, a single-threaded game running on a multi core system never takes advantage of multi-threading, what happens is that the OS keeps switching from one core to the other.
yes and no.
The game will only use one core at a time if compiled without SMP support.
However it is not entirely correct that it does not gain from running on a multi-core system. A system usually has a lot of background programs running and they can be moved to other cores and will not steal resources from the program.
Multi-threaded programs are extremely hard to test as they may crash undeterministic. Even if you run a program flawless for years an upgrade to the Operating system priority system might break the program.
Multi-threaded programs can have lots of problems that single threaded programs does not care about such as race conditions and deadlocks. The way to avert race conditions is to design the program to be multithreaded from the beginning that is too late for id tech3. I don't remember what part of ioquake3 that has problems. It might be the sound system or input system that has
possible race conditions.
Most multi-core systems are able to run OpenArena at full speed on one core, so SMP does usually not give any performance gain. If you have an old dual processor system like 2*500 MHz Pention III you might gain something but those setups are rare.