I don't think anyone is arguing that you do, the high-frequency sinusoid he described has a peak-to-peak amplitude of 2 * UINT6_MAX (although it appears as though the intention was a sinusoid with a p-p amplitude of UINT6_MAX) which on its own (in terms of the integer numbers given in his example) be represented in a 6-bit system. This isn't really relevant though, because that signal would be 0dBFS in a 6-bit system. The higher 10 bits are far from unused by the sinusoid in a 16 bit system, as they allow the amplitude of the sinusoid relative to the other sinusoid to remain unchanged. The little stairstep digitally-sampled sinusoid picture might look a little more rough for the "6-bit" sinusoid than the "10-bit" sinusoid but that's A)kind a pathological case and B)not at all representative of what gets sent to the amplifier after the DAC. (Think about the spectrum of all those little "stairsteps" and what happens to them once they are sent across a shunt capacitor...) anyways
I guess it doesn't matter to human ears with a well-mastered 16 bits, but the video linked in the OP explains that typically dithering noise is shaped (toward frequencies we're less attuned to). The models used to lossy-compress also typically put more noise in some of the higher freqs.
