If your core beliefs are so repugnant to the average person that they can be used as 'gotcha' questions, there's probably something wrong with your running for public office.
Reminds me of when The Daily Show had a Sarah Palin supporter on. Samantha Bee was interviewing the supporter and says "Don't you hate when the media reads verbatim quotes or shows unedited video and Governor Palin sounds stupid?". "Yes, the media is so biased against her."
First off, I am in no way implying that Sarah Palin would have made a good V.P.
But, taking a stupid quote that a politician said and using that as evidence they would be a bad (insert office position here) because of it, is kind of ridiculous when you realize they are being recorded pretty much 24-7. I have said plenty of stupid things in my life.
For example, Obama said the following "I've now been in 57 states I think one left to go." at a campaign event in Beaverton, Oregon, May 9, 2008. Does that make him unfit for office? No, he just said something stupid.