Children are generally able to handle much more freedom than modern parents give them credit for. Usually a bad result when you first try granting more freedom is due to a lack of practice, out of rebellion more than anything else. Just because a child wants control over her own body doesn't mean she won't tend to make similar choices to the ones you would make for her.
My almost 5 year-old daughter inherited some of my night owl genes, so she chooses to stay up later, but that's okay because she can usually sleep as late as she needs. When she's ready, she goes to sleep completely without complaint. My 10 year-old daughter who has approximately the mental capacity of a 3 year-old due to her cerebral palsy, actually comes and tells us she's ready for bed.
My 7 year-old son is somewhat different. His impulses drive him to stay up all night if we don't intervene, despite how tired he gets, so we tend to be more authoritarian with him. We don't like to do that, but find it necessary due to reasons I'll detail later.
Children have difficulties with freedom in two main areas:
- Considering medium and long-term consequences of their choices.
- Considering how their choices affect others.
The first reason is why I consider the job of a parent to be mostly converting long term consequences into short term ones. Kids need hygiene of their bodies and their environment. They need a certain amount of sleep in order to be healthy and happy. However, there's no obvious short term benefit to attending to those needs.
The way we handled those long term needs while still giving our kids as much freedom as possible is we made a list (with pictures) of all the things that need to be part of their bedtime routine, but we let them choose the order it's done, and give them wide leeway in how long it takes.
If your daughter still needs help with part of her bedtime routine, like brushing her hair, it's okay to impose more restrictions on those parts, because those parts affect you too. Children often realize intuitively, and others can be taught, that involving others necessitates some compromises. In other words, it's easier for them to accept some loss of control in those areas.
We keep the time reasonable with a consequence. After their bedtime routine, we have a family quiet time where we have a religious message and a family prayer, then if there's time we watch something funny like AFV. The kids really enjoy all of it. If they take too long with their bedtime routine, then they miss out on part or all of that family time.
The reason I feel comfortable imposing a consequence like that is because of the second difficulty with freedom. Making other people wait for family time is a choice that affects other people. Not letting parents have some quiet time after the kids' bedtime is a choice that affects other people, which is why I insist my daughter leave us alone after bedtime even though I don't insist she go to sleep right away.
That's also the reason we make my son actually lie down and try to sleep even when we don't force the other kids. The extremity of what he would choose for himself throws off the entire family's schedule the next day, and the resultant grumpiness throws off the entire family's mood the next day. We can handle a child sleeping an hour longer. We can't handle a hyperactive child being tired all day. If he could handle staying up all night without affecting the rest of the family, we would let him do it.
A lot of families have more strict morning routines than we do. They have to be to school or work at a certain time. That pretty much necessitates everyone going to bed at a certain time. That's an okay freedom to restrict on your children, because their choice affects others. It also happens to restrict the adults' freedom, but for some reason a lot of parents don't think of it that way.
However, you can still negotiate some leeway, and find a way to say yes. "Okay, you can stay up for an hour if you don't interrupt my quiet time, and if you can show me it won't be any more difficult to get you ready to leave in the morning." One of two things will happen: you will prove them wrong or they will prove you wrong. If you prove them wrong, they'll usually be much more willing to abide by the rules, which is worth the price of one or two difficult mornings. If they prove you wrong, you've both learned something wonderful and will be happier for it.
You asked about the benefit of giving a 3 year-old more control over her bedtime routine. The benefit is people are much more willing to do things they have more autonomy over. You'll have fewer power struggles and still be getting most of what you want. Also, I haven't parented teens yet, but I've heard from others that giving their children more freedom earlier eased the teen power struggles later on. Ironically, the best way to control children is often trusting them to control themselves.