PC: “I believe there is a concept that it is permissible to eat in someone’s house if they are shomer shabbos (correct me if I’m wrong)”
That would generally be true because there is a presumption of a Chezkas Kashrus. If I didn’t know that you eat food without a Hechsher, I would not suspect so and need not even ask. However, once I am aware that your standards are radically different than mine, I could no longer eat at your house. Your definition of the word “Kosher” is substantively at odds with mine.