Three-phase has two main reasons to exist:
Driving N. Tesla's polyphase induction motors used throughout industry.
Reducing the total cost of metal in cross-country power lines: w/single-phase lines, more metal would be needed to transfer the same rate of kilowatts.
You're right: lighting as well as AC motors will briefly turn off at 120 times per second (that's for USA 60Hz line frequency.) For this reason, metal-vapor streetlights and older inductor-ballast fluorescent tubes have significant "hum" modulation in their light output. This strobe effect can be very visible when your eyes sweep across strings of LED-based xmas lights. The hum is greatly reduced with incandescent lamps, since the filament doesn't cool down significantly during the millisecond-long low point in the 60Hz wave.
And you'll probably hear a 120Hz sound when using high-speed carbon-brush motors under high mechanical loads, such as electric mixers and carpentry routers. These high-RPM products can't use AC induction motors which are naturally limited to 1800RPM by the 60Hz drive frequency. The 120Hz variation ends up in the torque output of high-speed brush motors.
Single-phase induction motors should have much less 120Hz mechanical buzz, since they're essentially 2-phase motors with magnetic field torque which rotates rather than oscillates. Either capacitance or inductance is used to give the motor a second electromagnet pole having shifted phase.