Robots may fill the gap. Really. It seems silly now, but give it twenty years.
The developed world may end up with a modest human population and a large robotic population. Asimov explored that idea in SF decades ago.
The humans may still think they're in charge. They won't be.
I don't see why there has to be a "gap" at all. The country had half as many people in 1950, and it got along fine. The world is different now, but I don't see why it takes 300 million people to run the country today.
If the US gets back to 150 million people, it will look different, but I don't see why it has to be any worse than it is now. And I can think of a lot of reasons why it might be better.
Get rid of Medicare and Social Security, and the transition would be fine. The problem will be in transferring more and more wealth from the working and giving it to the non working, resulting in an everlasting loss of quality of life for the working (absent sufficient advances in automation).
The humans may still think they're in charge. They won't be.