Whenever I hear stories about this I always wonder if I am getting all the facts.
Although I can acknowledge there is some "good ole boy" mentality still hanging around, I don't know if race (or gender) is that much of a barrier anymore these days.
Usually, if you're good, you'll get moved up, regardless of skin color or which bathroom you use. Usually when the truth comes out, there is some other reason(s) advancement doesn't happen and while it could be of cultural origin that affects the person's personality/work ethic/management skill, it's not something that isn't fixable.
To me, if you are really as good as you say you are, they would match the competing offer and if you really want to move up as much as you say you do, then if they don't, you go to that other company and move up there. Unless you are okay with being where you are at due to stability or other factors... and if that's the case, I don't think you should be too concerned because you chose that.
I'm in a similar position where I think my salary at another company is probably close to double but at the same time, I am in a position of stability and being able to choose what I want to do so I take that into more consideration than the dollar value. I can complain I'm not being paid enough... but then I'm also not having to work insane hours, be under more scrutiny and not having the flexibility to set my own schedule and tasks. It's always greener but then you think about the fact that you can post on TI a million times and that makes it all better.