This isn't a good comparison. The technological ability for us to have flying cars has long since been there - VTOL vehicles have existed for some time now. We could easily create drones that could carry people, etc.
But flying cars have never been practical because of the human element and the lack of physical constraints that could readily be placed on their control. Plus, people realized that as ugly as a highway is, sky highways are a hell of a lot worse. The idea became less and less popular, and now there's no real demand for anything of the sort.
There's plenty of demand for precious metals. And the technology for asteroid mining, even largely automated, already exists. It's a matter of driving the costs down enough to make if profitable, or waiting until supply drys up enough that even if the costs remain relatively stable that it will still make money.
We already see a lot of progress and promise in launch costs being driven down. There's ambition within these companies to expand farther and do more. The technology is getting to the point where there's little question on whether it will happen, barring some sort of extinction of modern society type event, it's just a matter of how soon.
I don't necessarily expect asteroid mining to happen on any large scale in my lifetime, but, I also wouldn't be surprised at all if it does. In 100-150 years? I'd be surprised if it hadn't.