I agree that fact and truth are often used interchangeably without confusion. People say things like “It’s a fact that the book is on the table,” or “It’s true the book is on the table,” and in everyday speech it’s unproblematic. But that doesn’t mean that the distinction is useless. For example, we also hear people say, “That’s not a fact, that’s just what he believes,” or “His statement is true because he gets the facts right.” In these latter two examples, fact and truth aren’t doing the same work. The distinction is the one I’ve pointed out, i.e., fact is what is the case, and a belief or assertion is true when it gets what is the case right. My point isn’t to correct ordinary usage. I’m pointing out distinctions that ordinary language already allows.
The same applies to justification. When I claim that justification is concerned with whether a belief is connected to truth in the right way, I’m not saying that there’s only one method or one philosophical procedure. I simply mean that not every connection between belief and truth is enough for knowledge claims.
Of course many of these concepts are problematic, which is precisely why they need clarification, especially in epistemology. So, I think you’re overstating the problem.