First of all, that was intended as a general statement, not an absolute description of every case. Experiments have been done on people to see if, for example, they stop being opposed to incest in fictional scenarios where the incest is stated outright to be harmless.

Before the scenario was presented, people offered utilitarian justifications for the incest taboo, but even when those were stripped away, they insisted that incest is still "just wrong". My point is that this is what generally happens when someone points out incoherency in a moral system. People generally switch to offering an axiomatic rationalization for their moral sentiments instead of a utilitarian one.

Also, I have to say:

I have changed my mind about my values due to noticing that my values were inconsistent.

Do you mean that you made a judgement elevating one value above another you had in cases where they conflict? Or do you mean you actually gained a new value? It seems like you must have used some sort of higher level value preference to make that meta-level moral judgement.