I read, listened, and hope learned a lot from the #solidarityisforwhitewomen conversation and what I gathered is that for those (rightfully) offended by mainstream white feminism's lack of concern over feminist issues outside the mainstream bubble, an apology is useless because it's self serving. The author here makes the same point. An apology, when it isn't asked for, is saying 'please still like me.' 'Pretty-girls' like the author, Lindsey, and admittedly myself expect that to work and be enough - because it always has. What's really vital is action. The 'doing better'. Not just saying, "I'll do better," as a an alternate plea for being liked, but being AWARE and actively making sure the offenses do not happen again, not under your watch.
I'm trying to abide by the 'shut up, listen, and learn' advice when it comes to this conversation but this is my understanding of the point of the author's piece.