It seems to me that other survey non-response may be uncorrelated with C once you condition on taking a long personality survey, especially if the personality survey doesn't allow nonresponse. (I seem to recall taking all of the optional surveys and considering the personality one the most boring. I don't know how much that generalizes to other people.)

Why would the strong correlation go away after adding a floor? That would simply restrict the range... if that were true, we'd expect to see a cutoff for all C scores but in fact we see plenty of very low C scores being reported.

The first way that comes to mind to gather information for this is to compare the nonresponse of people who supplied personality scores and people who didn't, but that isn't a full test unless you can come up with another way to link the nonresponse to C.

Yes. You'd expect, by definition, that people who answered the personality questions would have fewer non-responses than the people who didn't... That's pretty obvious and true:

R> lwc <- subset(lw, !is.na(as.integer(as.character(BigFiveC))))
R> missing_answers1 <- apply(lwc, 1, function(x) sum(sapply(x, function(y) is.na(y) || as.character(y)==" ")))
R> lwnc <- subset(lw, is.na(as.integer(as.character(BigFiveC))))
R> missing_answers2 <- apply(lwnc, 1, function(x) sum(sapply(x, function(y) is.na(y) || as.character(y)==" ")))
R> t.test(missing_answers1, missing_answers2)
Welch Two Sample t-test
data: missing_answers1 and missing_answers2
t = -25.19, df = 806.8, p-value < 2.2e-16
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
-18.77 -16.05
sample estimates:
mean of x mean of y
9.719 27.129