If you want to raise your kids well, stop treating them like they are a rare Chinese vase and they’ll be just fine.
Generally, this is advised against to minimize chances of the child contracting a virus or bacteria while there are no good treatment options in the first few months of life. Also, putting them in a vehicle increases their risk of being in a collision, which has the same problem of having few good remedies, so playing it safe is considered OK.
Helicopter parenting is a problem, for sure, but that isn't any more american than obesity is.
It's parenting based on fear, a reality of being in the information age and less fortunately the age of social media.
It seems to me that children can thrive under a wide variety of parenting styles. I consider that my own upbringing was a bit "rough around the edges", but with age I respect and understand my parents more and more.
In retrospect, I wouldn't have it any other way. I respect my parents a lot more than I used to.
"No one ever walked into my psychology practice because their parents made them too emotionally independent."
There are some parents out there who continually martyr themselves because they think it makes them a good person (I get 4 hours of sleep a night to make sure my kids have everything! I'm awesome!) but are in fact overinvolved and setting a bad example for the kid by not taking care of themselves. Anecdotally I've found most of my married friends who are/were hesitant about kids were raised by such parents.
It's healthy for the kid to see that the parent needs some enforced level of time to themselves, short of neglect of course. The precise balance is up to the individual.