My experience is with private schools, too (mostly LACs, mostly selective or moderately selective), and there was surprising variation re: how seriously they tracked DI. Some listed it as “important” and some listed it as “not considered.” This seemed unrelated to how selective the LAC was, whether they offered interviews, etc. You just have to look at the CDS.
Interviews are not usually evaluative because the quality of the interviewers is so inconsistent – whether they are student interviewers or alums, they receive bare minimum training, and some are better conversationalists than others, and their reaction to the students they interview is highly subjective. So colleges typically say that they use interviews as yet another source of information, but they will never make or break your application (I think some are different, like Harvard or MIT, but I’m not talking about those kinds of schools), and they are one of many ways to demonstrate interests for colleges that track that. Obviously it can only help to have a good interview, but it won’t hurt you to have a mediocre one (or none at all). I’m sure if you reveal something about yourself that’s truly awful, that’s a different story.