[kroltan]

How do we add more colours (besides just picking a random colour, which wouldn't be helpful)?

By sampling the signal more often ("multi-sample anti aliasing"), also known as increasing the resampling rate, then representing that with a wider bit depth (not just 1 bit "yes/no", but multiple bits forming a color/opacity), since we do have more than 1 bit per pixel that can be used already.

I'll give it to you that this is "anti aliasing", not "not having aliasing in the first place", but the Fourier argument above is the reason why in computer graphics we practically always have to "settle for" AA instead.