It makes absolutely no difference to the integrity of the game world whether you randomly roll the presence of that beard or declare it's presence or absence based on anything else- player wishes, the phase of the moon, whatever. None of this matters the slightest whit to the game world.
One reason I'd make a random roll to see if the NPC was bearded (1-3 no, 4-6 yes) is because it helps to remove my bias from the game. Beards aren't a big source of bias for me, but a while back I realized that most of the NPCs I was putting into the game were men. Now I roll randomly for sex and I think it's helped out my game.
I wouldn't always roll - I may know that, in this town, there's a taboo against being clean-shaven (or the opposite). Well, I might roll 1d6 with a 6 indicating clean-shaven (and then come up with some kind of reason why this guy has no beard).
I do recall a moment where a PC ranger came across some treasure while hunting/trapping in the wilderness - he found the treasure with a "stunning success" on the foraging roll. I said he found a rabbit warren and, inside that warren they had some gems. Both the player and I wondered what the rabbits were doing with gems; I said that on a 6 on 1d6 the rabbits could talk. I don't remember clearly, but I think the player may have suggested something like that.
As you can see there is still some bias there (I had the rabbits talk instead of having them pets of a ranger, druid, or some other NPC/monster), but it feels like a different kind of bias, though I'm not sure what the difference is.
But I don't follow your point under the "players" interpretation either, because in this context the only player behaviour is asking the question "Does the NPC have a beard?" Answering yes to that question is not "being biased by player behaviour". It is responding affirmatively rather than negatively to a question from a player.
I think that the difference here is that, when a player gets an idea for a plan, the cards always seem to fall into place for them. The player may be asking only because they are thinking about a plan, not simply gathering more information about the game world. Randomly rolling for these minor things can give the player a sense of internal game-world consistency - things don't always fall into place for them.
By saying no, what the GM in effect does is shift the focus of play away from what the player wanted - namely, finding out whether or not the plan with the disguise works - to making the players come up with another plan. I don't see how this helps the game. (Unless you're playing Tomb of Horrors style, in which the point of the game is for the players to accommodate their plan to the pre-written backstory.) And I don't see how the player's perception of the gameworld is going to be altered, and destroyed, by having his/her hope satisfied. The player clearly wants the gameworld to be a certain way. And s/he knows that the GM has the power to make it that way, and (unless s/he is very confused about the difference between reality and authored fictions) s/he also knows that it is by means of the exercise of that power that it will or won't be made that way.
Two points:
1. A lot of D&D can be summed up as what you describe as "Tomb of Horrors" play. Without intricate backstory, which would be very difficult to come up with for everything the PCs might encounter, the DM needs another way to make impartial judgements. That's why I use random rolls (a simple 1d6 roll works well most of the time, I find).
2. Rolling randomly takes the responsibility off of the DM (to a point - the DM still has to determine when to roll and how to assign the odds). Doing so creates a less adversarial/mother-may-I game, I think.
Do these same players get upset when the GM decides that the patron walks into the inn and asks their PCs to go on the MacGuffin-fetching mission without first rolling a reaction check to find out whether or not the patron likes the look of the PCs?
Deciding without a random check that the patron hires the PCs and not some NPCs, or deciding that its the PCs who are walking past the plot-hook event rather than rolling for that on a table, seem to me to be far bigger instances of the gameworld reality shaping itself to external considerations, than deciding that an NPC has a beard because a player is hoping so. Yet they are the stock-in-trade of every GM everywhere since time immemorial. No one's game has the PC's live boring, uneventful, poverty-ridden lives simply because random content generation gave all the plot hooks to the NPCs!
This is the kind of thing I would do. There are features of my game that mitigate this, though. The game world has enough adventure in it that the PCs don't need to wait for NPCs to offer quests; the PCs are free to react to the NPC's actions in any way they want (say, following an NPC adventuring party them to the dungeon entrance and jumping them when and if they come out); a zero-sum power struggle between characters (NPC - NPC and NPC - PC) in the game world; and an XP system that rewards player-driven goals.
