I remember watching this the other day and wondering if this would come up. I didn't think the gag was over the line but I did think it was unnecessary, having Colbert say a bunch of stereotypical things really quickly when the real issue was about a racial slur did sort of overshadow the point when I watched it. I walked away remembering feeling conflicted about the gag more than the real issue.
Additionally, while the tweet by CC was stupid and out of context, the show didn't properly give context to the origin of the character Colbert was doing, it just showed the old clip of him doing this character. I hadn't seen the old episode so I thought it was Colbert satirizing his earlier episodes, like maybe he too had done something stupid in the past and was mocking it, reading this article and now knowing that the character had a point back then makes me think it was short-sighted to bring up this old character without giving his origin proper context. They could have simply done the same gag of him starting a foundation with a racial slur in it and I felt it would have had a better effect since I was so conflicted with Colbert doing the whole stereotype thing.
Picking blame between CC/Colbert and CC's twitter/Colbert's twitter seems like a silly distraction, I can personally see how someone might get offended even with context by the gag Colbert made. I think focusing on explaining the context and then apologizing if people were offended given context would have been better and might have calmed this story down. However, overall, if you want to get a TV show cancelled, you shouldn't use Twitter because Twitter is often part of the problem with racial insensitivity, exampled by the fact that it was used to attack/insult this person who might have had a legitimate complaint about the gag. Also, a hashtag shouldn't be a key term to a complaint, you are putting your movement for sensitivity among the likes of FML and YOLO and that doesn't help your case.