I was born something like 6 weeks after the EDD, which was obviously an error (even if I was "late" I really doubt I was 6 weeks "late," lol).
And I, too, think of this with fresh perspective since my mother was never pressured or made to feel afraid about it. I don't know what happened, but I'm guessing that when the EDD came and went they eventually said, "hmm, we must have gotten the dates wrong, oh well." My mom went into labor on her own and had me vaginally and drug-free within 2 or 3 hours after she noticed contractions. My mom never really even made a big deal about that part of the birth story (how "late" I was) when I was growing up, she mentioned it a few times but no big thing.