Like others have said, it really depends on the context. “Exquisite” offers an alternative without suggesting any hint of fragility. Elizabeth’s exquisite discernment allowed her to detect the slightest trace of insincerity. As for craftsmanship or any endeavors marked by excellent skills, I would stick with "delicate". With a delicate touch, the poetess inscribes a new page upon the canvas of imagination.

Delicate can be used in any of these senses! I think it depends on who you're speaking to. I would use delicate to describe artwork or something that was made using a very fine process. But if someone mentioned that a person was "delicate" I'd definitely think they meant "sensitive". As you mentioned, though, this usage sounds a bit old-fashioned and literary, and most native speakers would probably use "sensitive" or even "touchy" to describe another person's personality, depending on nuance.

In poetry or in a romantic way, i think "delicate" could be a beautiful word. In a business environment or among aquaintances, there is a chance it can offend.