That sort of cavalier attitude is what directly leads to intermarriage.
Hey, she thinks she’s Jewish, she self-identifies as Jewish, so what’s the big deal. Let’s treat her as a Jew and if later a real Jewish man married this gentile woman, why sweat too much about it.
This is exactly what Rav Elyashiv was concerned about, leading him to seriously consider establishing a Yichus Registry to document who is and who isn’t truly a Jew.