tl; dr: the stronger you are, the more likely it is that you can cheat in this way and not get caught.
Check out this position:
[FEN "r2q1rk1/5p1p/p2p1bp1/4pn2/2B5/7R/1PPQ1PPP/2B2R1K w - - 0 1"]
This position is from Kasparov-Anand, Las Palmas, 1996. It's White's turn to move, and Kasparov went into a deep think. You might want to think about the position yourself, although if you are not a strong grandmaster, you probably won't see anything special.
At this point Kasparov went into a deep think. Jan Timman started to speculate whether White couldn’t play the very forceful 20.g4. Kasparov’s second Juri Dokhoian immediately confirmed: “That’s what he’s looking at!” Yuri understands Kasparov’s thinking better than anyone else in the world.
Meanwhile White had played 20.Bd5. The game lasted six hours, Anand defended very tenaciously and at around 10 p.m., much to the disappointment of Kasparov, a draw was agreed.
When he left the stage Garry spotted me and walked straight over. “I couldn't win it, could I, Fred?” he asked, with a troubled look on his face. It was a bit shocking: the world champion and best player of all times consulting a chess amateur, asking for an evaluation of the game he has just spent six hours on!
Naturally Garry wasn't asking me, he was asking Fritz. He knew I would have been following the game with the computer. “Yes, you had a win, Garry. With 20.g4!” My answer vexed him deeply. “But I saw that! It didn't work. How does it work? Show me.” He and Anand listened in horror while Juri dictated the critical lines. All of this was captured on video and published in ChessBase Magazine 56 (Feb 1997).
Part five of the series continues:
Most top grandmasters understand all too well how computers can affect the outcome of a game. In contrast to an amateur playing 600 points above his true strength, for whom the computer must dictate practically every move of the game, a strong grandmasters requires only occasional assistance to improve his performance dramatically. There are usually a few critical positions in which a player must decide whether a promising plan can work, or whether it is tactically flawed.
If we return to the example of the Las Palmas game given in part four, Kasparov’s second knew what he was looking at and actually uncovered the solution before the move was played. All that Kasparov needed was one bit of information: “Yes”. He didn’t need to get the message “20.g4! wins” but simply “There is a win” or even, in this specific case “The move we know you are looking at works!” That would have been enough to decide the game.
So yes: if you are a strong grandmaster, then having access to a computer telling you there is something in the position will result in a huge elo boost. You don't even need to know what the move is, you just need to know there is something. Furthermore this kind of cheating will not be easy to detect, since for almost all moves of the game you will do just fine on your own.
On the other hand, if you are an amateur, then you will need help from the computer much more often, and furthermore you'll need specific help. Look again at the position: would knowing that White has a win have helped you at your level? As you mentioned, detecting cheating is by matching one's moves against the computer's. Naturally, once you have to consult the computer more often, cheating becomes easier to detect.
Edit: Quoting from this source on the kind of cheating you propose:
Faced with a complex calculation, a player could sneak their smartphone into the bathroom for one move and cheat for only a single critical position. Former World Champion Viswanathan Anand said that one bit per game, one yes-no answer about whether a sacrifice is sound, could be worth 150 rating points.
"I think this is a reliable estimate," says Regan. "An isolated move is almost uncatchable using my regular methods."
[Regan is an academic who's developing methods to catch cheaters.]
But Regan has ideas on how to catch this kind of cheating. They are untested, but he's confident they'll work. See the source for more.