|How would a company feel about you if you broke three interview appointments? They would figure that you are either uninterested or a flake, or perhaps an uninterested flake. Is there any reason for you to think differently of them?|
You can go ahead and call the company for yet another interview, and they will probably schedule it for you. Why shouldn't they? They can always break it. The fact is that if they were truly apologetic and equally interested in you, they would have already set up another interview date or they would be calling you right now to do so.
In this labor market, with such a scarcity of applicants in so many fields, a company that treats a potential employee with such disregard is waving a gigantic red flag. And, in your case, they have waved three of them.
The best step is to continue your job search and focus on companies that truly respond favorably to you, rather than to a company that "seemed to respond favorably" to you. If that original company sets up yet another interview for you, and something better comes along, you know what to do.