I would argue that it does. More abstract concepts often (and especially in this case) elucidate a concept with wider application. General relativity gives us insight into a possible model of our universe. Godel's incompleteness theorem gives us insight into all complex axiomatic systems in all possible universes. The difference is definitely there and an argument can be made (up to senses of aesthetics) that one is better than the other.

No. Incompleteness is only true for logical systems that are sufficiently like the one we've got in our universe.

Regardless, your basic argument continues to be that "more general => more profound." Which is quite simply not a true principle.