For example: If a person tells another person, "You're racist" and the other person responds, "No, I'm not racist. But I just don't like to see Black and White people dating because it's not right", then the other person responds back, "Thanks for making my point."
How about self-defeating? From Lexico:
self-defeating: (of an action) preventing rather than achieving a desired result; futile.
The person in your example is making a self-defeating argument, thereby inadvertently making your point for you.
One phrase which could describe such a person is
give (oneself) away
To reveal or make known—usually unintentionally—something about oneself, such as one's actions, intentions, ideas, guilt, etc.
- Margaret had been embezzling money from the corporate account for years, but she finally gave herself away when she claimed her two-month trip to Paris as a business expense. (freedictionary).
Aside: This may be beside the point, but I would also describe such a person as guilty. If you are not guilty, you have nothing to give away.