Annie: Problem Child?

I seriously have a few problems with Annie, and I'm not sure if my grievances are legitimate or if I'm just bad. My biggest problem, which bleeds off into the other smaller issues I have with her, is quite honestly the combination of her passive and her ultimate. Annie's stun feels like it lasts quite a while by the time she hits level 6, and with her ulti being instantaneous if you're in range, you're out of luck. And then you combine that with the rest of her incredibly reliable and high-burst kit she can 100-0 most squishies before the stun even expires. To give a point of reference, her kit has only TWO sources of damage with AP ratios under 80%; her Molten Shield, which essentially gives her a Thornmail with an AP ratio, and the DoT from Tibbers.

Annie just doesn't feel like she has any weaknesses. Her Q does a lot of damage, is auto-target, and on-kill restores its mana cost AND halves its cooldown. Combined with Doran's Ring or ANY mana/mana regen items and she is not the slightest bit mana-gated. She has good range in spite of her overall reliability, and with her Molten Shield she can make herself quite durable, mitigating the typical squishiness of burst mages. She has 335 movement speed, making her faster than other champions with less reliable kits.

Yes, Annie has no mobility spells or gapclosers outside of Flash, I am well aware of this. Honestly it doesn't really help, give her a weakness, because if she has you in range there is no dodging ANYTHING except her Incinerate, and even that's questionable. It feels similar to Ryze in all honesty. "Bad scaling" really doesn't seem to be in effect here; she certainly doesn't get easier to deal with the farther along a game gets, especially when she gets fed. I'm not sure if Banshee's Veil existing counts as a weakness, because it's not an innate vulnerability in her kit but an item you HAVE to sink gold into.

Annie's an introductory character, I get that. She's supposed to be easy to play and understand for people new to the game. But that doesn't excuse just how frustrating she is to play against, to the point that she's a high-priority ban for me. So really, am I just missing something? Is there a means of properly dealing with her in solo queue?