Many children do go to relatives to live. It's all too common today. But for the most part, children are better off and safer with their parents even in less than ideal homes.
The child does not have the choice though. They cannot decide to go to relatives, a friends, or whatever/wherever.
If their parent beats them, is just rude and inattentive, or of course rapes them or something, they have no guarantee that they can simply go live with somebody else. Sure, they can report it, and go to CPS, but that's awful, and that relies on them being confident enough to tell and not afraid.
That's helped by being given explicit rights over who is your caregiver.
And you don't seem to see the love that other people have for children in general and their own children, or that they're not treated like animals or property, and they're not disempowered or voiceless.
I'm not saying all children are treated as such, however that is how many are (In America that is). I'd say, arguably, most are.
Some people really love their kids, some people love all kids, but from my observations a huge portion of society doesn't like their kids all that much. They enjoy moments with them, they feel obligated to them, but it's rarely ever a deep "I'd do anything for them and do it gladly" type of love.
You must live outside the USA or live in a big very liberal city in the USA and in an upper class neighborhood. Otherwise you should see this, at least unless you treat the kids as they do, and thus see it as perfectly adequate.
Keep in mind most parents (in the usa) admit to hitting their child at least a few times, and the majority even admit to doing so when they were a young toddler or baby.
Most parents (in the usa) also command their children and talk down to them any time they do something wrong.
Most parents (in the usa) also yell at their children when they're annoyed or mad at something the child did.
What would you call that?
I'd call it unloving and abusive.
but it would be wrong to let them exercise them before they're truly able to
They're perfectly able to in most scenarios. If they don't like tennis and want gymnastics, don't sign them up for tennis lessons because you like tennis. It's really that simple.
Unless a child has no clue about the topic (in which case educate them) than they can more than adequately make good choices for them, as they know what they want generally.
As I said, it's not true love. How do you define love, by the way?
How isn't it true love? The only people who even compare to their love for kids is people who support gentle parenting, attachment parentings, baby-wearing, bed-sharing, etc, which is a very small portion of people, mostly hippie vegan mothers.
I define love as the genuine willingness to give your everything to somebody and truly do anything for them. To be selfless in regards to them, to put them before you no matter what. To truly care about their feelings and happiness, even before yours.
If you love somebody that shouldn't be something that hurts you, if it's a healthy relationship, as you should derive pleasure from doing what your love enjoys and desires.
Few people truly love in this world, and an insanely small percentage truly love their children, yet alone others/all children.