Let me put it on the line - LFR and Flexible raid sizes are the most important raiding experiences currently available in World of Warcraft, and the upcoming Mythic 20 person raid difficulty is an atavism, barely even an appendix, that only a vanishing few players will experience when it is current. It exists for a sense of achievement and prestige that only a few players really have the time for anymore, and every year, that group of players gets smaller.
The reason for this is simple - as Tom Chilton put it, the demographic is getting older over time. People like me who played for the raid game back in classic are older. They have jobs, kids, schedules that don't permit the kind of time investment hard modes currently demand, the kind Mythic will demand. And it's not that you can't do cutting edge raiding in, say, six hours a week. I'm not arguing that you will have to put in 20 hours a week to do Mythic. I'm arguing that even scheduling one or two nights a week and being there reliably is actually really hard when you have other commitments that can often demand your time on a moment's notice - in essence. being able to go when you want/need to raid instead of when the group is scheduled to go is a huge boon to that aging demographic. For all the elitism, all the sneering, and all the slurs directed at the LFR player base, the feature allows people who love raiding but who can no longer commit to scheduled WoW play a place to do it.
You can ask if this is healthy for the game as a whole - whether or not your answer is yes or no, though, there is no escaping this simple fact. WoW is a decade old. Many of us playing it have been here for years now. Even players who started in Wrath or Cataclysm have now been playing for years. This is an aging game with aging players, this is the reality of the situation. And this means that more adaptive raiding solutions are going to keep presenting themselves.
The addition of Flex in Mists of Pandaria was a huge game changer for people who veer between these extremes. I know quite a few people who love playing, who have multiple max level alts, but who simply can't commit to raid group schedules or find raids that start at times they can make - either they have work that doesn't end until after the guild starts, or they have evenings with family that take up that time, or so on. We could list reasons all day. The point being, these players aren't suddenly bad at WoW. These are players who got realm first Yogg-0 kills. These are players who were beating Heroic Lich King before the Cataclysm systems patch. They simply don't have the ability to commit to the game the way they once did. That doesn't make them terrible players.
Now, I won't pretend there aren't lots of terrible players in LFR. There are. That's because there are terrible players in all levels of WoW, and those terrible players are still players. It's true that guilds serve to winnow out some of those terrible players. One of the reasons I loved raiding with my guild was that I knew I didn't have to worry about people completely ignoring mechanics or not even having their gear optimized, forgetting to pick talents, etc etc. One of the things you have to accept with the more democratized raiding experiences like LFR and cross-realm flex pugs (or using OQueue to run normal Siege) is that you'll get some unskilled players, some jerks, sometimes both. This is the curse of the player who raids as he or she can. But this doesn't change the fact that LFR and flex get a bad rap from players who misunderstand what they're for.
Neither are meant to replace more structured, more time consuming levels of raiding. When Mythic raiding is live in Warlords, it will be a niche that a very small amount of players get to participate in. That's fine. That's what it is for. LFR and normal raiding (what today are LFR and flex) will be for players who require more give in their gaming - the players who can only play for an hour or so after the kids are in bed but before they have to get to bed because they have work in the morning, the player who really can only devote weekend time to raiding, the player who never knows if she or he will be able to make tonight's raid because work keeps having crisis after crisis that needs to be put out. These players' still enjoy the game, still love to get to see the really rather astonishing raids Blizzard puts together. As the demographic ages, more and more of us will become these players.
This is the future of WoW. LFR and flexible options will only grow in importance. There's a reason all non-Mythic raiding will be flexible in Warlords.