Black footballers were extremely rare in the 70s and even when Barnes played still a small minority. So its not surprising that from a small pool of players there are only a few black managers. Its only since the 90s that football has become colour-blind for players. Its inevitable there will be a delay for managers (please finish the post before jumping on this).
So yes, black managers are under represented, but its mainly a sign of racism in the past. I think we are in the transition period now. With the large pool of black footballers, there will soon be a lot more black managers. There will also be change in the board room where the old generation who were brought up when black footballers were rare are replaced by those who expect to see black footballers in every team.
I believe that because of the NFL experience. In the 80s it was still thought that you couldn't have a black quarterback, let alone black coaches. Yet slowly black quarterbacks made their mark (Williams, Cunningham, Moon) and now there is nothing unusual about a black quarterback. The same with coaches, there were very few through the 90s, but now are much more commonplace. The NFL did introduce policies forcing teams to interview black coaches for coaching jobs, which probably sped up the change that was happening anyway.
However, while I believe the change will occur inevitably, we shouldn't be complacent. It must be frustrating for the generation of potential black managers who lose out. While I don't think Barnes or Ince were good managers, its a fair point about white managers getting many opportunities. I also can't help looking back at all the managerial changes we had while Chris Hughton was an assistant.