The DCs come of a DC-by-level chart, with fairly precise guidance about which column to pick from. The players initiate skill use, because they are the ones who declare actions: see eg DMG pp 74-75:
Sometimes, a player tells you, “I want to make a Diplomacy check to convince the duke that helping us is in his best interest.” That’s great—the player has told you what she’s doing and what skill she’s using to do it. Other times, a player will say, “I want to make a Diplomacy check.” In such a case, prompt the player to give more information about how the character is using that skill. Sometimes, characters do the opposite: “I want to scare the duke into helping us.” It’s up to you, then, to decide which skill the character is using and call for the appropriate check. . . .
In skill challenges, players will come up with uses for skills that you didn’t expect to play a role. Try not to say no. . . . If a player asks, “Can I use Diplomacy?” you should ask what exactly the character might be doing to help the party survive in the uninhabited sandy wastes by using that skill.
But the fundamental difference from B/X or 5e is as
@Campbell has described: the skill challenge framework is a binding resolution framework that generates consequences on the way through and yields finality of outcome.
Still not seeing a difference.
5E has DCs by difficulty as well. It also works the same way.
As you mention... "I want to make a Diplomacy Check" and "Can I use Diplomacy" are incomplete player actions and are rejected at my table. I would ask for more information about what the player wants to accomplish. I would require their actual in-game action. I would also require their intent and their assumptions of success. "I want to trick the Duke" isn't enough, for example. I need to know how they want to trick the Duke and for what purpose/outcome?
B/X doesn't have skills. But, I'd approach these things in the same manner. I would get the action, intent, and assumed outcome and apply the rules. Often getting the aforementioned information clarifies the matter. In the case of interacting with the Duke, that would be a Reaction roll. Modified by any situational bonus based on the approach of the player and the in-game situation. If there are no rules that apply, I would make a ruling. I already have a very good understanding of the action, so it is relatively trivial to do so.
I can achieve the same results in adjudicating player actions in 5E and in B/X as you can with skill checks and skill challenges in 4E.
My ruling would look something like this: "You can attempt <intended action>, if you succeed <what you assume happens> if you fail <something else happens>". The outcomes of success and failure will be clearly defined for the player because I know what the player intends and expects. You have the same possibility of consequences and the same finality.
I could ask for a skill check or not. It would depend on the situation... I could just rule that the player automatically succeeds or automatically fails based on the situation at hand. I can make a ruling tailored to the situation at hand without being constrained to satisfy some game mechanic construct.
I handle all of my player rolls this way. Every time I ask for a roll, I need their intent and assumptions and I state their success and fail states. If I can't think of a success and/or a failure for a possible roll, then the roll is not needed to begin with.
I prefer the versatility of making rulings.
In Prince Valiant there is a general system for simple resolution, and a general system for complex resolution if that is what is desired/appropriate (margin of failure is deducted from the pool - whoever reaches zero in their pool loses). Any skill can oppose any other, because of the uniform framework.
As I posted not far above, B/x and even moreso AD&D don't bear comparison in this respect. In B/X, for instance, how do I resolve a fighter's attempt to stage an ambush by hiding? How does that interact with the thief's Hide in Shadows ability? How does Move Silently relate to the surprise rules? What about the rules for evasion of pursuit? Even in Moldvay's example of "there's always a chance", the rules for falling damage are disregarded. When is that meant to happen? Can a 1st level player make the same appeal to a small chance of stabbing an ogre in the vitals and so winning the fight, without having to engage the to hit and damage rules?
5e isn't quite as lacking in integration of its systems as B/X or AD&D, but it has similar issues. In a running race, for instance, what benefit does a fighter get if his/her player is prepared to expend a use of Action Surge? What happens if one character wants to woo another by singing a song - is it Performance check vs Wisdom save?
Etc.
In B/X, if a fighter wanted to stage an ambush, the player of the fighter would describe what actions his character will take to set up the ambush based on the in-game situation and the DM would make a ruling if it would be successful or not. The thief would do the same thing. If the thief wants to use their Hide in Shadows, they may do so. The use or not of Hide in Shadows doesn't change the situation. The thief can prepare an ambush in the same manner as the fighter without consideration of their thief skills.
Surprise is an abstraction. Surprise rolls aren't something players actually do. Players don't say: "I want to roll Surprise against the orcs". Surprise is a roll made by the DM as a part of setting the context for an encounter. It is used when there is a lack of in-game cues to determine if whether one group is aware of another or not.
"There's always a chance" is also not used in the manner you are suggesting. It is a catch-all rule for handling a character doing something. It serves the same function in B/X, as an ability check or a skill check would in 4E or 5E. The rule also states nothing about disregarding falling damage or automatically winning a fight.
Now, if you have a player who is falling and wants to attempt to brace their fall by grabbing onto an awning or use the wall to push himself towards landing on a bail of hay, such that they can lessen the impact and reduce damage, you may want to invoke this rule. You can ask for a Dexterity or Strength roll. (Note the description I used: the player has to describe what action their character wants to take, their intention for taking the action, and their assumption of the outcome). I could not make this ruling without that information so it has to be provided.
In the other situations described above... handling a fighter using action surge in a running race, or wooing someone with song, these are outlier areas that are left to be handled by the GM making a ruling.
In a running race, a GM can look at the in-game situation regarding the race and make a ruling. Does the fighter want to use Action Surge to give them a quick boost of speed? Or are they tapping into their reserves in a marathon? The description of what the player wants to do and their intent needs to be known. The GM can rule based on that intent.
Wooing someone with song would be exactly as you suggest. In 5E it would be a Charisma (Performance) opposed check vs Wisdom. In B/X it would be a Reaction Roll. Before rolling you need to know how the player is doing this ('what song are they singing?"), you also need to know what the player wants out of it (what favor they want from the NPC). Then you can determine and state what happens if you succeed or fail. The GM determines difficulty based on the guidance of the rules and based on their understanding of the NPC that they created and presents the success/fail states. The player then chooses to make the roll.
I see the role of the GM as the arbiter of the game. Every game is different and every situation that develops in a game is unique. It is unique because of the in-game situation, the context of the adventure/module, and even the nature of the people at the table. The best person to determine the outcome of events at the table is the GM. They are the ones with "boots on the ground', so to speak. They have the best information as far as how a situation can be successfully and satisfactorily resolved.
