Yes, women that make up stories are making the situation worse for other women. The more that they make up stories, the less seriously the police take it when the next woman calls.
If the woman in the Mixon case had been a man, people would have laughed and said he deserved to get his arse kicked. Even if the man was a tiny little dude, nobody would have sympathy for him if he initiated the contact.
Mixon probably could have done more publicly to show remorse, but it's difficult to do it without making yourself look guilty.
I'm fairly certain that Greg Hardy's girlfriend set him up because she had a plan to get paid. Regardless, he handled the situation terribly. If he had hired a PR firm, he could have done many things to show that he understands that DV is a serious issue. For the money he could have donated, he could have gotten some group that is against DV to publicly shake hands with him and acknowledge him as a great guy for donating a million to their group/cause.