Here's why ADCs are "mandatory"
[EDIT] I apologize for not responding to each individual comment below. I'm highlighting and addressing general trends I see in edits at the end of the main post.
I see this "argument" thrown around a lot, and most of the time it reads as "Riot has a bias towards Marksmen" with the assumption that if Riot simply guts all Marksmen characters or removes them from the game, the problem would be solved. It won't.
League is an objective based game, built around optimizing the tools at your disposal. I won't go into the whole explanation about resource allocation among team members, but essentially it comes down to this: objectives (towers, inhibs, dragon/baron/herald) have a lot of HP compared to champions. Moreover, towers/inhibs are primarily damaged through auto attacks (there are some spells that deal damage to structures, but it's typically minimal compared to auto attacks). Therefore, the fastest way to take down a tower/inhib is with a champion that deals a ton of damage through auto attacks. To optimize this, you're going to want to allocate as many of your resources as possible to maximizing your auto attack damage. This means building a lot of damage items, and fewer defensive items (or none). Because you have fewer defensive items, you're going to be really squishy, meaning having another advantage like range helps you not get destroyed when you go up to attack a tower.
This is why Marksmen have been in every single meta. This is how the game is innately structured. Riot could literally delete every single designated ADC in the game, and guess what? People are going to start playing TF/Kennen/any-ranged-champion-whose-auto-attack-doesn't-suck in the ADC role. If Riot deletes every single ranged champion in the game, players will just move to Tryndamere or whatever's left over that auto attacks a lot and takes towers quickly, but the game will still stall out since you won't have an opportunity to get significant damage on towers unless you force your enemy to back/kill them. At the core, there's nothing that makes an ADC unique besides the fact that they're ranged and they can auto attack (yes, there are some particular champs who have unique scaling, but this is a result of Riot accepting this basic reality of the game structure and working with it). This is why Riot even went through the Marksman update to try to differentiate the class. The vast majority of ADCs damage comes from items, something that any champion can buy, which is why you can play things like Fiddlesticks ADC; he satisfies both requirements (1. is ranged, 2. can auto attack).
One last point: there's the classic MMO trinity of Tank-Healer-DPS. League doesn't quite follow this formula, but the fact still remains that so long as there are things with HP, you need damage to take that HP. There's this pervasive misconception that auto attacks don't have any cost. They do: it's time. ADCs trade time for consistent damage. (EDIT: Just to clarify, I mean time in the animation and cooldown between auto attacks; time in the sense that they deal damage over time, as oppose to burst damage, with some exceptions; and time in the sense that it takes in-game time for the ADC to scale, which, consequently, reduces the "time" aspect in the first two costs).
Bottom line is: unless Riot fundamentally changes their game structure, the concept of an ADC isn't going anywhere, regardless of what happens to the individual champions.
EDIT: I see ~2 common themes in the comments, both of which are very important to understanding why ADCs are an ever-present part of the game, and both of which I (intentionally) did not address above. The first is the issue of resource allocation among team members, which I mention above, but chose not to address because it doesn't go directly to why ADCs are in the game at all, but mostly to why ADCs are in the duo lane. I won't go into it here, because there are plenty of comments below that explain it well, but I do agree it has a contributory effect on why ADCs are a core member of a team (ie. if someone has to take a hit in experience gains, ADC is a good choice because they're mostly item reliant, less level reliant).
The second theme is a ranged ADC's advantage in teamfighting, especially over melee ADCs (ie. Tryndamere, Yi) and many other champions. Again, very important contributory effect as to why ADCs are on virtually every team, but not absolutely vital. I chose to focus on ADCs ability to take objectives because you can win a game without teamfighting (it's rare, but possible), but can't win a game without killing the nexus (unless the other team surrenders or something, but that's not exactly a reliable way to win a game).
EDIT2: I see a lot of people commenting something along the lines of "champion X could fulfill the same DPS role if Riot balanced the game differently." Which goes hand-in-hand with my point above that if Riot deleted all designated ADCs, people would just move to the next best champion that fulfills the role. Like I said, there's nothing really unique about the concept of an ADC besides that they're ranged and they deal auto attack damage. The designated ADCs just happen to be the champions that are currently best suited for the role. It doesn't change the fact that the role is a core part of the game.