As someone with thick hair, I feel your pain. I'm sorry you are feeling so frustrated. I completely understand though. I just chopped mine off (10 inches) and thinned out recently.
Firstly, let me tell you that thick hair is not a curse. In fact, most people consider it a blessing.
Secondly, you should know that you have options. You mentioned cutting it short and then straightening it. How short are you thinking about cutting?
Your hair is overwhelming you due to it's sheer size and how time consuming it is. It doesn't have to be that way. You can reduce both size and time while still being curly. Trust me. How "easy" a hair style is only how "easy" you make it.
Think long term about what you think you want to do with your hair. This is a very personal question... What are you going to do in the humidity? Swimming? Rain? Sweat? All of these things destroy flat ironed hair. (Not to mention the severe damage that is most likely going to destroy your curls.) Are you going to be okay with this?
I think natural hair is the easiest and I have never flat ironed more than a few times a year. Even with my hair THIS short, I hate straightening. It's extremely time consuming and meticulous. I personally can't picture myself flat ironing regularly so I can't give you any tips about it.
If you're stuck in a rut, I think you need a change but not a drastic one. Try getting a new cut (though not a SUPER short one) and see how you feel. You would be amazed and what some long layers can do in terms of reducing bulk. If you like it, you can always go shorter.
I hope that helps. Let me know what you think about all of this.