I can see where you might have got the idea that “leading” means “advantage”, but that’s not the case here. In this case, “leading edge” is an idiom that means “advanced.” I found a good explanation of this idiom at this website:
“Cutting-edge” and “leading-edge” are both adjectives used (especially in product promotional materials and corporate “mission statements”) to mean “At the forefront or most advanced stage of development; highly innovative or pioneering.” Both are also used as nouns (without the hyphens) to mean that highly-advanced state of awesomeness (“Professors, commonly assumed to be on the leading edge of thought…” Fortune, 1983). “Cutting-edge” and “leading-edge” are most often used in the context of technology (computers, cell phones, flying cars, etc.), and only rarely do we read of “cutting edge quilts” or “leading-edge pot roast recipes.”
In your sentence, the writer is employing some wordplay there. A researcher may be on the leading edge of science, and professor David Roitstein wants the CalArts program to stay on the leading edge of music technology, but this hotel is on the leading edge of edgy. Taken literally, that seems like a redundant expression: We can have a leading-edge hotel, or we can have an edgy hotel – why say both? But I think the author is onto something; it’s a “punny” way to say that the hotel is ultra-edgy, or on the "extreme edge of edginess." Given that Paris is such as trendy location, I’m assuming this is intended to be quite high praise for the hotel being reviewed.
As for why it’s acceptable to use an adjective as the object, that’s an astute observation that I think many native speakers would have missed.
I looked up the expression “at the leading edge of innovative” online; I found many hits, but the word innovative was always modifying a trailing noun (as in, “at the leading edge of innovative design”). So the rules of grammar are indeed being bent here, but it’s a good example to demonstrate the flexibility of English.
Because of the deliberate wordplay, I think edgy can function informally as a grammatical noun in your sentence – even though it’s normally used as an adjective. The phrase “at the leading edge of edginess” might seem like a more grammatical way to say it, but the problem with that is we associate the word edginess with nervousness, while the word edgy is more closely associated with trendy.