Earlier quoted context omitted.
The linked Verge article has some strange editorialization in it. I find it confusing, because I'm not sure what they're getting at. > Systrom then immediately peaced out for a few months of paternity leave. > With Systrom on leave, Facebook began testing location tracking and a disgusting hamburger button inside Instagram, offending Systrom’s delicate sensibilities. Is the author saying that Systrom was being to sen…
The primary definition of delicate: fine in texture, quality, construction, etc.: a delicate lace collar. I read it as a compliment. If he had said "delicate ego" it'd be a different story.
As for people, I don't think our culture values fragility in an individual. Conversely, we definitely do value resilience. I'm not sure of a case where you could refer to a physical or mental aspect of someone as delicate or fragile (when it's relative to the norm where it's not) and it comes out positive.