So this is specifically the Wizard's spotlight time?c
Not sure where you get that from. He simply does not wish the other characters, and by extension their players, to be aware of his subterfuge and sabotage. The mystery is part of his enjoyment of the game. As well, as we see from the interplay between the players, they do not always agree as to what would be the most fun for playing out the game, and the wizard’s player has selected this approach to exercise some narrative control.
He trusts the GM to ensure that all the players get spotlight time in accordance with their preferences in the course of this exploration, later events or both.
Does the wizard propose that this is a time machine and make a roll to have that affirmed in play? Or does he ask if it's a time machine and make a roll, the outcome of which is controlled by the GM?
Perhaps the machine was initially established to be a time machine (a fact never shared with the other players) or perhaps this is the first the wizard has thought of it, and is using it to vary the scenario by removing the time pressures which might impede the other players from agreeing to his desire to explore this new environment. In the latter case, the GM sees no opposition to this added aspect, so he says Yes rather than rolling the dice.
Good news? Who says what the Wizard's knowledge check indicates the lighthouse material is useful? The Wizard? The GM? The group?
The wizard wants to find a way to sabotage the conveyance in such a way as to require exploration of the city. The GM has decided to roll the dice, rather than just say yes, as there is a conflict between the wishes of the wizard and those of other players. Between them, the GM and player determine that a success means he has a viable approach, which he and the GM flesh out between them.
This ties back to who created the lighthouse - one method of play would see the players create the lighthouse and then tie it's usefulness (or not) to their own goals in some way. Another sees the GM narrate the lighthouse and the players 'ask questions' about it (the answers to which give the GM the required tools to guide the players along). The two don't mix very well.
The GM threw out the description. The lighthouse was a simple reason the city would be visible from further away, and he has not established its use. However, either he or the wizard player then suggests that the magic to operate this device might use a common component in magical artifacts which the conveyance also uses, facilitating the sabotage, which that stellar die roll indicated the wizard has clearly determined how to carry out.
Why not simply say 'I'm sabotaging the machine in a way I can fix' and then engage the resolution mechanics.
This is basically what he has done. However, the player is interested in having his efforts fit seamlessly into the milieu the GM has offered up, so he asks the GM for his input on what may work. The player asked whether this was within his capabilities, within the rules of the game. The GM said yes, it is possible, but not automatic success.
Why does the player need the GMs permission to do this? Who's in control here?
The game is a collaborative effort, so control is shared between the GM and the players.
Why not say this in the open so all the players can enjoy the subterfuge (even if their characters do not). Do these players not trust each other to make the game more fun?
Perhaps they also find mystery and subterfuge fun, and find removal of all mystery detracts from the fun. I’ve commented in the past that I have no desire to have a player say at the table “My character has been betrayed several times in the past and, as a result, he has become secretive and distrustful of others. As a result, he is not telling your characters his true identity, and that is why he always keeps his face concealed.” I find it much more engaging to allow character background and personality to emerge through actual play than through a written narrative providing the player with information the character lacks. I did not need to know Vader was Luke’s father from the first appearance of the characters on the screen either, nor did I need Vader’s claim confirmed during the Cloud City duel. It is more interesting to find out as the characters do.
The players trust each other to make the game more fun to an extent sufficient that full disclosure is not required in order to maintain that trust. Just like Hussar and Pemerton trust the players will use their authority to fast forward a scene wisely and in the best interests of the game. However, at least two players (Fighter and Cleric) do not extend that trust to the GM, in the same manner Hussar, from his comments, does not. The Wizard player, however, extends that same trust to the GM.
I'm interested in the fact that the one time in this whole example it's made explicit that a player is attempting to take some sort of mild initiative they feel the need to a) ask the GMs permission and b) do it in secret.
a) He is engaging the GM to adjudicate success and failure, a primary responsibility of GM’s in most game systems.
b) The players like mysteries, so they often do things in secret. They do not wish to be burdened with the chore of keeping in-character and out of character knowledge segregated, and prefer to have only that knowledge their characters have, to the extent possible. They have charged their GM with responsibility for keeping OOC knowledge out of play.
The hints are that the GM is in complete control here, irrespective of whether it meets the group's approval or not. However, as my questions illustrate in most cases there's simply no indication of what's happening between the people at the table.
That is the conclusion Fighter and Cleric have reached, certainly. It is what I believe most players would believe with the initial information available to them (pre-spoiler). It seems crystal clear, however, that two players want to skip the city exploration, one very much desires to play it out, and the fourth is fine with whatever plays out. The GM’s desires are not expressed explicitly – he has no voice at the table. However, the players make their own assumptions, right or wrong. What did you assume before reading the spoiler?