That's just saying that the heat is dissipating at the bottom of the heating element, in this case the kettle. The heat is moved by convection by the air molecules, then conduction by the kettle, then convection again by the water.

But for the water to reach the boiling temperature, it needs the underside of the kettle to reach its boiling temperature too. At sea level that would be 99.8C / ~100C, but at higher altitudes, it's less.

If the kettle did not reach the boiling temperature, the first law of thermodynamics would not be upheld: the total energy of an isolated system is constant aka conversation of energy. The water must reach a boiling temperature and the only item between the flame and the water is the kettle. If the kettle didn't reach the boiling temperature of water, the water would need to get the higher energy from another heat source. Without such an additional heat source, it would thus be getting it from the void, which would allow the creation of perpetuum mobile.

Also, the paper cup doesn't burn because it is thin enough to convect the heat to the water and thus not reach its combustion temperature (233C). It still reaches the boiling temperature of water for the reasons states above.