I understand that we perceive the sky as blue and not violet because (1) sunlight has more blue than violet in it (see here), and (2) our eyes are more sensitive to blue than to violet.

However, I can't reconcile that with the reality that there's plenty of indigo and violet left in the rainbow, which is after all a breakdown of sunlight.

Whether I give more weight to (1) or to (2), I find a contradiction with the fact that we do in fact have plenty of violet:

If one were to argue point 1, "there isn't enough violet in sunlight", then one could counter with "but I can still see plenty violet in the rainbow".

If one were to argue point 2, "we're not sensitive enough to violet", then one could counter with a similar answer: we seem to be sensitive enough to violet as seen in the rainbow.

(If anything, the fact that the atmosphere prefers to scatter violet over blue should give extra points to violet in the color of the sky versus the rainbow, which is borne of less scattering—am I right here? The color of the sky is made from scattering from many, many rays that passed us overhead. The rainbow is made from much fewer rays that interact with water droplets. Less scattering.)

Yet, I can't deny the reality of what colors I'm perceiving. So, what is the physics that is eluding me in this regard?