We already live in that world, and it's been that way for centuries.
If all the automobile designers and mechanical engineers were to vanish today, do you think lay people would be able to hop into the driver's seat and take over the job? The same is true for nearly every industry, and always has been.
The only reason we've come this far as a civilization is because we record what we learn and pass the knowledge on to future generations of people who are interested in that knowledge.
Not everyone has the same interests, and barring an apocalypse, no one needs to know it all at once. Programming is no different.
Sure, things like reading, computer literacy, math, biology, basic problem solving, etc. are important base skills for everyone to know. Once those foundations are in place, delving further is a matter of preference, and not everyone should be pushed to learn computer science any more than they should be pushed to learn quantum mechanics.