I remember seeing several recent games where 2600+ players played Leningrad Dutch but at 2700+ level they tend to prefere more solid lines like Slavs and Semi Slavs. If they plays more dynamically they tend to prefere KID or Grunfeld.

As far as I can see the top players hardly ever play the dutch.
They throw in Jänisch, Aljechin and the Dragon but never an Ilyin- Zhenevsky.

What is the concrete reason?  Any clear path to advantage for white?

I regularly play the dutch, have studied the theory and don't understand what is the problem. Leningrad, Stonewall, Dutch-Indian, Illyn-Zhenevsky... OK you may have to play the first moves more accurately than other openings, but aside that I'd say they are as good as any other 1.d4 defence.

For example according to Williams book, only one critical line leads to a small advantage in IZ with 7. ...Ne4.In my experience from some standard games and many blitz games, very few white players know any deeper dutch theory so you usually end up with a good position.

However, the following game may be a nice example of what happens when you trust current theory and play stronger players:

[Event "It"]
[Site "Bunratty (Ireland)"]
[Date "2008.2.23"]
[Round "3"]
[White "Svidler Peter (RUS)"]
[Black "Williams Simon (ENG)"]
[Result "1-0"]
[Eco "A90"]
[Annotator ""]
[Source ""]

1.d4 e6 2.c4 f5 3.g3 Nf6 4.Bg2 Bb4+ 5.Nd2 a5 6.Ngf3 b6 7.a3 Be7
8.Ne5 Ra7 9.e4 Bb7 10.d5 fxe4 11.Nxe4 exd5 12.Nxf6+ Bxf6 13.O-O
d4 14.Bxb7 Rxb7 15.Qh5+ g6 16.Nxg6 hxg6 17.Qxg6+ Kf8 18.Re1 d5
19.Re6 Nd7 20.Bf4 1-0

7. a3! is a new and very clever move where black cannot stop e4 and d5, each dutch player's nightmare (7.. Bxd2 8. Nxd2!). Previously 7.Ne5 has been the main move. 

I guess Svidler checked Williams book for some nice holes