Annie is a really simple champion with high rewards for understanding her binary kit. It's not about people being "caught out" by her, it's more about the fact that she can instantly turn around any teamfights/skirmishes with her flash + R. Even without flash, a good Annie will be positioned such that her Tibbers will turn the fight. It's a lot of damage, a respectable stun, and it's not possible to dodge it unless you pre-emptively flash.
Annie is to mid/support what Warwick is to the jungle: a very simple champion with very little counterplay (largely due to their simplicity itself). That's probably why people complain about her. Warwick has purposefully been kept under the nerf hammer because Riot doesn't want him dominating the more "complex" and "fun" junglers.
As for mobility, blinks and dashes have become a festering rash on the game. It's very difficult to play any immobile champions anymore, and if an immobile champion is within the meta it's because their kit is heavily overloaded on utility or damage or both. League has always had a probably with overtly mobile champions, right from day 1. When the developers decided to remove turn rates from the game (that is, how fast your champion can turn around), it created a problem with ranged champions being able to easily kite melee champions. They then put a bunch of blinks and burst damage in the game: the assassin age. Since that became a problem, they put a bunch of CC and tankiness in the game without removing enough of the damage: the bruiser meta. Since bruisers became a problem, they decided that megatanks were needed to answer: the tank meta. Now we're slowly emerging from the tank meta and going into another cycle of the bruiser meta.
The kits just get more and more overloaded to answer for the piling up of the past metas' sins. Of course, this doesn't fix anything and just makes a mess of a game. Then you have champions like Annie who have weathered the ages due to good fortune with the nerf bat and binary kits with high rewards and versatile applications.
This has been League of Legends History 101. Remember to do your homework; there will be a quiz next week.