The traditional 機 was the wooden loom — the most intricate machine ancient China knew, whose trigger mechanism also fired crossbows. From the loom's critical moment came 机's second life: the pivotal instant, the opportunity.
手机 hand-machine is your phone; 飞机 flying-machine, the airplane; 机场 the airport; 司机 the driver. And 机会 is opportunity — the machine-moment when things could go either way. One character runs China's hardware and its luck.
Wood 木 loom, the first MACHINE — and every MOMENT of opportunity it weaves.
危机 'danger-machine' is crisis. The motivational claim that it means 'danger + opportunity' bends the etymology — the 机 here is the critical pivot-point — but the speech-circuit loves it anyway.