I definitely would not go against her wishes and cut it. I think that's unfair to her and would most definitely be deflating for her...
On the other hand, you definitely could use some different strategies for handling her hair. First, there are hundreds of different kinds of tie-backs, I find it hard to fathom that you can't find one that will hold her hair in a pony tail. They even make elastics (fabric coated) that have some rubbery substance on them, that are "no-slip" kinds... if you haven't tried them, I would... There are also silicone hair bands (pony tail holders) that hold tightly, with little to no slippage. I have both, so I speak from experience.
What are you using to comb out her hair after washings? I'd say that if she has hair that is extra tangle prone, to use a leave-in conditioner, or a detangling spray at the least, maybe both, working the conditioner through the length of her hair. Also be sure to use regular conditioner after shampooing, if you aren't already, as this will definitely help with reducing tangles and making the hair easier to comb when wet. I also highly suggest a VERY wide toothed comb for combing wet hair. Brushes will wreak havoc on wet hair, especially hair that is tangle-prone. My wide toothed comb has teeth that are about 3 inches long, with large spaces between them... and it is wonderful. And ALWAYS comb from the bottom up... start at the bottom, remove those tangles, and work your way up to the scalp. This is the easiest on the hair, and the scalp!
On blow drying, why bother? Unless you are going out in frigid weather, and her hair is wet, why not just let it air dry? How often does she have to wash it? Kids that age usually do not need to wash it very often, maybe every several days? If so, do it in the evening, and let it air dry... OR, do as a close family friend did with her little girl. Her little girl had a TON of hair, and she'd wash it in the evening, then let it air dry a little, then before bed, she'd put it in one braid in the back, and let her sleep that way. In the morning, she'd have nice waves, it was at least mostly dry, and it wasn't a tangled mess. The braid kept it the hair in place.
It seems to me that there are lots of pretty easy ways to resolve some of these hair issues, and I agree, this shouldn't be such a struggle. I've had varying lengths of long hair my entire life, so I speak from experience with long hair.
Lastly, a thought to keep in mind is that it will only be a short while until she's really old enough, and preferring to do her hair herself... so this won't be your battle for long...