However, the culture of helicoptering kids everywhere (and hovering nearby, trying to control and risk-manage everything) is spreading to Europe more and more.

Sadly. This is one of the negative effects of having mostly-foreign entertainment media, and I’d hope that the effect could be countered soon.

Entertainment is one thing, but I think far from the only factor in spreading this.

I tend to think that US is just somewhat ahead in most developments, and trends then come to Europe. In addition to near-hysteric control of what children do, we'll have gated communities in fear of crime, we'll have politically correct "safe spaces" in universities, and whatnot current US phenomena.