Yes, you only need that many samples given you know a maximum frequency. But to actually perform the conversion from discrete samples to continuous wave takes infinitely many steps. The number a given DAC actually does varies. Some do hundreds, some thousands, a couple rare (very expensive) ones do 100k or more.

That's what I'm saying. Increase the number of samples in the audio itself, and the DAC can get closer to the original.

Except it's not. The video shows that. In the real world, the DAC doesn't immediately jump from code N to N+1; it slides (the "slew rate" of the opamp). More bits just makes the difference between codes N and N+1 smaller; i.e. where it slides to is more accurate to the original. The "stair step" or "straight line" view you get from programs like Audacity is a lie.

A final low pass filter (tuned to remove anything above half the sample rate) allows the accurate reconstruction of the original analogue waveform from the sampled digital data. ( The vertical 'stair step' of sampled data is very high frequency information which gets removed, 'bending' the output back into it's original shape). The video linked above is well worth a watch.