Not that I'm an expert in traveling with Jewleries but I think the bag looks nice. I would try to click it into the key ring hook most bags have (or try to find a place where you can tie it to) - then the screeners can't take it out just like this (unless they cut the cord which would require quite some quick action)

Regarding checking valuables, that should be a no brainer: Be it "Thiefrow", JFK or many other airports - I heard so many horror stories I personally wouldn't risk it.