1.e4 d6 2.d4 e5 3.dxe5 dxe5 4.Qxd8 Kxd8

I would think that if this were ever played in a GM-level game where white was in a must-win position, that he would maintain the tension with 3.Nf3

I've played black's position a couple of times in club games. I like it from a psychological point of view. Perhaps good to play on Friday night weekend tournament against a tactically good and promising 12 year old who is way past his bedtime and may succumb to a Sits-Krieg approach.

Digressing, 1d4 d6 2c4 e5 3de de 4 Qxd8 + is better for black. (Hole at d4 and bolthole for K at c7) This knowledge can be used wrt to the mentioned Philidor starting position in later maneuvers.

OF COURSE from black's point of view you can get this position a lot from games and you could become something of a national champion with this variation, because is is very NARROW going into it. Most people open 1e4 or d4 and after 1...d6 most games would continue with the other central pawn going forward two squares

OK I should have qualified it by giving more moves and saying usually it's the case that there is an hole on d4 if white tries to assert his God given right to central dominance.

Here's a game I played as black:

It's difficult to say that black had anything before white played 26.Be2?

I believe white has no advantage after Qxd8.

The poor statistics for black are mainly because after 5.Bc4 many black players play 5...Be6, after which 6.Bxe6 gives black an isolated double pawn. It might not be enough of a weakness to win it, but it is one weakness, and if black creates another weakness --> two weak points are usually enough for a win.

Instead of 5...Be6, black is fine after the natural 5...f6, but it is suprisingly not moved so often, which explains why white scores so well after Qxd8.

I do play this position as black and I believe that black has an easier game than white, resulting often in a considerable time advantage in an equal endgame position. If white continues O-O at some point, black can hope for an endgame advantage because his king is closer to the middle in the endgame, but of course white does not need to play O-O, but might prefer Ke1-e2 instead, where the white king is usually placed comfortable. The move O-O-O might be more troublesome for white, because the d-file isnt so important (no way white rooks will make it to rank 6 or 7), and thus the absence of a Ra1 is troublesome after black starts pushing a7-a5-a4 backed up with a Ra8.

All this indicates why this is difficult to play for white, but strictly speaking he has no theoretical disadvantage. It just is easy for black to get a stable position, whereas white has the opportunity to play many good looking but weak moves. 

Black is rather playing s.th. like Nbd7/c6/Kc7/b6/Bc5/a5/Nh6-f7 and if BxBc5 then Nxc5 with Be6 next. Of course it depends on white´s moves which moves black plays first, but black has a clear shaped plan where to place his pieces, resulting in saving a lot of time. 

There are two tactics black needs to take care of:

after c6, Nb8-d7, Kc7 and Bf8-c5, white can sometimes play RxNd7+ with Be3xBc5 next, winning two light pieces for the rook. The usual way to avoid this is playing b7-b6 (which guards the Bc5).

The other tactic white often has is the exchange sacrifice: RxBd6 (or sometimes RxNd6), which might win 2 pieces for the rook, too. As black: Watch out for the d6 square!