Actually English is not my first language so I may have used some terms not in the proper way.
If pitch is just about singing the note perfectly right than I think it's not the same as singing on key.
I mean, you can still sing perfect notes but off key. Singing on key acappella is about having
in your mind the chords and singing the right intervals between notes. I don't really know how to improve
it, but I think that playing an instrument can be beneficial because you put more attention on
notes, chords and intervals than you do while singing. But I'm just guessing.
But again, acappella is difficult even for some pro singers, I've seen the italian version
of "the xfactor" and in an interview the winner sang cappella and she started
in one key and ended half tone higher. But she is a very good singer. What I want to say is
that going off key acappella is not important if you don't do that while singing along music.