The red shift applies to distant stars too. The important point is that even in the extreme scenario where all stars last forever and you wait an infinite amount of time for light to reach you, you will still never see a fully illuminated sky because the stars are receding much faster than the light propagates due to metric expansion. Saying the sky is dark because 1) the Universe is young or 2) stars are short-lived is just false. The Universe gets darker and darker as it expands, despite there being "more time for light to reach us".