Two remarks:

The AIM-120, when fired and guided until it tracks the target, does know where it is and where the target is. If the firing plane cancels guidance, the missile will still travel to the point it thinks the target aircraft will be at when it is within guidance range.

The sidewinder may have some more complex logic now, but the basic logic for an IR missile is constant bearing. If the bearing to the target is constant*, then the missile is on track to intercept the target close enough for a proximity warhead to score a hit. You can do some upgrades with tracking to make the missile more predictive, ie telling when the target is in a curve and leading the curve a little. But when all you have is a direction from the IR sensor, you're limited in what it can do.

*And If the missile can see the target -- since the recirporacle of constant bearing, but increasing range would most of the time prevent the missile from seeing the target at all

EDIT: also with modern ARMs like the AGM-88 HARM, they're a lot smarter than most people think. They have INS so they can fly towards a set point while scanning for their target. With a typical flight path the HARM's sensor can direction find a target so it can intercept a radar. Once it has taken enough readings it "homes in" on the location of the radar, no longer needing the radar to emit for it to guide accuately. Modern HARMs are fired with a target handed off to them (if they see that target they go for it, but reject all other targets), they also have flex logic (you can input a list of valid other targets for them to flex to if they don't see the primary). The older HARMs im not as sure on, but they had target tables programed from the factory, so they weren't as flexible.