Dame Joan Sutherland did not have perfect pitch and yet anyone who ever heard her sing the mad scene from Lucia di Lammermoor would certainly doubt she had not. Her pitch, alongside the flute part, was perfect to listen to. From this we can deduct that not having perfect pitch may be no obstacle to being a good singer.
I am minded to think of identical male twins I knew when I was in the church choir many many moons ago. One had the most gorgeous boy soprano voice: his twin brother was typically what we would call tone deaf and seldom hit a note proper. We had to put up with the one in order to enjoy the other and their Mother insisted they do everything together! We accepted that as a given but no-one was keen to sit close to the one who could not hit a note as it used to put you off what you were doing yourself. In the end the choirmaster put him in among the bass and alsto singers where he got drowned out affectively with no cruel intent.