Why can't I access songs that I have already purchased on iTunes? It shows that I "Purchased" it buy the song is "greyed" out.

Have you check in Music Library?

When I search for an album in iTunes Store that I had purchased, I also see purchased.

When I click my Library I see the album:

Try to -> Redownload music in the iTunes Store app on your iPhone, iPad, or iPod touch

Redownload music - Apple Support