I also prefer "endgame study." That's my personal priority. Keres, Shereshevky, and Nunn especially.
But lots of folks in this thread apparently "insist" that you will never reach the endgame, playing against them. And they might be right. 
So one might study the royal game in "reverse." First endgames, then middlegames, and openings last. But lots of folks would probably disagree with this suggestion.
Openings and endgames can be learned systematically and played relatively fast.
Middlegames are full of "suprises," both tactical and strategic, and it's very, very, easy to make "unnecessary" pawn moves that effectively weaken your postion in the middlegame.
But it's entirely possible to play "safe," narrow openings, and to play "defensively" during the middlegame--focusing on "safe exchanges" ala the Genede Nesis "Tactical Chess Exchanges" and "Exchanging to Win in the Endgame," books."
And if you are willing to simplify middlegame positions, via "safe piece exchanges," then you will certainly reach an endgame. Preferably a level one, at least. Then you are free to "play for a win."
But if you play into the teeth of sharp, mainline, classical openings, then of course you might not ever get into the endgame. As the three games listed above amply demonstrate.
But if you're playing the white pieces, then surely (using the suggestions above) you can probably force you're opponent (via simplifying exchanges) into a "level endgame." Just use the Edgar Mednis book (@1982) "From the Opening into the Endgame," and blast you way (with white) into the endgame.
And "endgame knowledge" will win games, BUT only if you'll willing to adjust your openings and middle game tactics to actually reach those endgames.
So don't take risks, and wait until you reach the endgame, before playing for a win. I seem to recall that Pal Benko was "greatly feared" for this type of play.
And as for what phases of the game to study most, isn't that mostly a personal preference? And the source of endless, mindless, blathering arguments, about what personal chess style is "the best" way to reach whatver chess level is being asserted.
In my experience [USCF A Class, five times, but never over 1900, OTB] chess players in the 90+ percentile (>1800 USCF) are largely obsessed with the game, and devote @3-4 weekends per month competing. Their wives are chess widows.
Personally, I belive any "serious student" of the game can make USCF B Class (1600-1799) in 1-2 years by studying just Tactics and Endgames and playing regularly, OTB. But to play consistently above 1800 USCF you'll need a different, "business model." And probably one that's adapted to your personal style and preferences.
So if 90 percent of active OTB tournament players (in the U.S.) never break 1800 USCF, then how high a rating can you get "without touching openings?
Probably 1800 USCF, if you are (at least) partly "obsessed" with the game. And lower still if you are not so obsessed.
I don't know anyone above USCF 1900 who isn't "obsessed with the game," or retired, or both. Indeed, lots of the retirees have preserved their ratings because that have stopped competing OTB. If they starting competing OTB, or G/15 or G/30 again, "the kids" would have them for lunch, and they know it.
Floors would keep them from going below a certain level.