Ruy Lopez, Morphy Defense, Exchange Variation 4...dxc6 5.d4 exd4

The general idea here is to get to the endgame as quickly as possible because black's doubled pawns will become a bigger liability in the endgame.

The position looks better for White to me, but Stockfish says that White has lost his advantage. At the amateur level, should this be considered a good opening for White?

I had heard that Bobby Fischer played that but looking through his games now, he played Ruy Lopez, Exchange, Gligoric variation:

Black has the bishop pair, that counterbalances the doubled pawns, thus it isn't the most common at the highest level. However, there have been some instances where the Exchange Ruy has been played at the highest level, often to force black to play actively when he doesn't want to or hasn't mentally prepared to do so. A notable example of this is Lasker's famous win over Capablanca in the 1914 St. Petersburg tournament:

The line with 5. d4 brought Lasker some successes but it got quickly discovered that with Bd7 and 0-0-0 Black's development and two bishops more than match White's superior structure (what we get there is not an endgame but a queenless middlegame). Then 5. Nc3 became the main line til Fischer's era when 5. 0-0 took over.

Yep, the bishop pair is Black's advantage - this is why the exchange is 'equal' and all three results are possible. Modern players tend to value the bishop pair slightly more than pawn structure, but that is not to say White does not have his play - I would say that it is easier for White to find good places to develop his pieces, whereas development for Black is somewhat slow and there are fewer decent setups. 

I'd say this early d4 variation of the Exchange is an opening where the player who understands the position better will thrive - it's not high on theory, nor really a tactical position, just a queen-less middlegame with plenty of scope for both players to pick the wrong plan. 

This is interesting, I don't think I've seen anything with ...Bd7 and ...0-0-0, I thought it was always the ...Bd6, ...Be6, ...f6, ...Ne7 setup. Maybe I've had it confused with 5. Nc3 all this time!

Most of the venom is taken out of the exchange variation.

5 Nc3 in the exchange is also interesting and it brought Alekhine some nice wins.

However, the delayed exchange brought Giri success, e.g. against Ding at the Yekaterinburg Candidates'.
1 e4 e5 2 Nf3 Nc6 3 Bb5 a6 4 Ba4 Nf6 5 O-O Be7 6 Bxc6
The apparent loss of tempo 4 Ba4 6 Bxc6 is explained by the misplacement of Nf6 and Be7. In the exchange black usually wants ...f6 and ...Bd6.

I don't know if it is just my own games but I have a theory: at the lower levels (say <1500) in the majority of games the knights are worth more than the bishops, because knight forks are missed more often than bishop forks. A single monster knight will often win a game in short order.

I wonder if there have been any studies to either back this theory or refute it?

#8
"The weaker the player the more terrible the Knight is to him, but as a player increases in strength the value of the Bishop becomes more evident to him, and of course there is, or should be, a corresponding decrease in his estimation of the value of the Knight as compared to the Bishop" - Capablanca

I think it depends on the handling of the position.  If the player with the bishops has long connected pawn chains, the knight is impotent.
If there are a lot of weak pawns and weak squares to attack them from dotted around, the knight is likely more useful.