Gilmoy: <Once: This site focusses on the end of a chess game.> Well, the PotD leans that way, if we only try to solve it from the diagram :) Hence I now try to solve an earlier puzzle-before-the-puzzle: <when did the winner see it?> N.B. This approach works in non-puzzle games, too.
<Put your bits on good squares.> But this is par; hence even doing this peters out against an opponent as sound as you. Super-GMs do this against each other, get perfectly fine development, and then struggle for 30 moves over ideas we still don't grok.
Conversely, a miniature already implies some fundamental unsoundness from at least one side. Let's restrict to the case of the mismatch, e.g. a GM simul, or a 2600 vs 1900 wherein we say "rating difference, uh huh". How does the pro overwhelm the weekender?
Clearly, <good squares> is a start; let's call that <Phase I>. Beyond that, I add a tier of thinking: <supports many plans> (or "matches many patterns", which is probably highly correlated -- and that could be an eureka item itself.)
- I borrow your suggestion of working backward. A winner-by-force finishes the game in what we may call <Phase III: Lock-on Autopilot> mode, where he's already calculated it to a win, and he's just skiing down the ply-tree. Spielmann skis as well as Alyekin.
- Assume you have all necessary firepower, i.e. a winning plan exists, but you must find it. How do you know to look? Here, I think good players stratify through <intuition, pattern recognition, experience>, or sometimes outright home (prep) cooking. You must know all the mates; you must evaluate endgames correctly (to cover all side branches); it helps to have seen hundreds of similar positions, attacks, sacs, and wins. A noob never sees it (on either side), a tyro thinks "there's gotta be something like THAT sac", a master thinks "X-Y 1907, also G-H 1923, you missed a draw". (Nakamura says "done that, what's for dinner", he's already in Phase III.)
- By definition, then, Phase II is when you <don't> have a win-by-force all calculated out (but you've completed Phase I). This seems to me to be exactly the newbie's "what now?" fog, through which we've all gone (and not all passed :) A general corollary is: <improve your pieces>, i.e. move from their good squares to(ward) better ones. But <good> by what?
I surmise that a master relies again on his experience/intuition: that <many successful attacks> flowed from a positional characteristic (c.f. Nf5 in Ruy/Spanish!!). Over a corpus of thousands of games, you can rank piece-outposts (and thus repositionings, tours, mini-plans) based roughly on the average outcome of games in which they occurred. (From other sports, sabermetrics and Billy Beane's Moneyball did this in baseball, and ESPN/NFL just did it for the NFL QB rating system.)
Hence we park Bs on diagonals, get Rs to the 7th, plant Ne5/Ne6/Nf5, Qg3/Qg6 and rook lifts, and push our Ps to 6 or 3. Call these <fertile outposts>, from which tactics flow -- e.g. the knee-jerk Ne5 always threatens Nxf7!!, which has bamboozled even an Aronian. Hence, I think what a GM does in 3 seconds during a simul is to judge: what would be useful across the <largest swath of possible game-continuations>? That's a race, too: if your opponent just lets you amass these advantages without booting them out (e.g. Breyer g6 to deny Nf5), then at some critical mass you reach your Phase III transition -- which see.
<In today's game, Alekhine whips up a killer tactic by move 14 because he puts his pieces on better squares than his opponent.> The Nf5-Bg5-Q*3 complex is thematic in Spanish (which is an experience-lesson). The key idea of deflecting g open when Q-sees-g is the sting in the theme. Even the thrust <8.d4>, enjoying the mobile recapture <9.Nxd4>, is straight out of Sicilian theory. Black went wrong by allowing this, and then not contesting Nf5: his corpus is smaller than Alekhine's corpus, and his danger-sense was more rudimentary than Alekhine's lockon-sense.
The upshot of all this seems to be: play through many games to broaden your corpus. Pure calculation solves the II-III phase transition, but it doesn't get you <to> it.