The problem is that it would be bad. Annie has one "job" as a mid/support and that is to be a huge threat to others thanks to her stun and Tibbers. If her stun "times out" or otherwise can be lost because she doesn't use it, she loses the utility of being able to choose when to fight.
And yes, it does make her very frustrating to face because a good Annie player can and will sit on a stun for as long as necessary during the laning phase when they are facing someone who isn't willing to risk getting hit by it. It gives her a huge advantage in that the pokes that would ordinarily kill her never happen.
There are solutions, one being get your jungler to come by as Annie isn't going to be able to stun 2 people at once unless they stand right next to each other.
I also would point out that it isn't "oppressive" for her to hold a stun. It's wise, especially as from what you are saying you became unwilling to face it. My last time out on Annie I ended up against a Riven in midlane. Riven tried to all in my one time at level 2 and I used my stun to burst her. After that Riven was unwilling to confront me directly and I was able to farm much more confidently. The truth was that I probably could not have burst her again from levels 3 through 5 as she had picked up some items that gave her the advantage, but the Riven player wasn't willing to take the risk and I was able to get ahead on CS, which helped my team the rest of the game. It was the psychology of having the stun that won the lane for me.
And that's what happened to you. You got so afraid of the stun that you wouldn't do anything. Remember that probably 85% of Annie's damage is in TIbbers (her ult). Her Q and W are not usually high powered until late game when she has a lot of AP items, so level 3 through 5 if you are willing to risk the stun you can often win the trade if you commit fully right from the start.