He is definitely saying "insight", it's just the way he talks. There is no home-grown American who would make a mistake like "he had no inside into." Now I don't know how y'all talk up in Seattle, but down here we actually have a similiar way of pronouncing "t" in this case - something that should be a "t" sounds more like a "d", especially before a vowel sound. I've recorded some sound clips of me talking normally:

1. We need some light in here!
http://www.freewebs.com/pravit/lightinhere.wav
2. Could you give me a glass of water?
http://www.freewebs.com/pravit/glassofwater.wav
3. I saw him dancing at a playground.
http://www.freewebs.com/pravit/dancing.wav
4. She's too arty for me.
http://www.freewebs.com/pravit/arty.wav
5. I saw a car accident at an intersection back there.
http://www.freewebs.com/pravit/sawacar.wav

Now, wouldn't you say I sound like a true good, God-fearin' 'merican?

Actually, I had fun doing this. Any more Russians(or Americans with weird accents) who would like me to record good ol' American pronunciation?

EDIT: JFK is John F. Kennedy. Another one is FDR(Franklin Delano Roosevelt).