What if countries had to compete for citizens?

To some degree, they do. Access to the world's best talent is competitive and it often goes to the highest bidder (in wages/quality of life/etc.) America and e.g., Albania, both presumably want at least a couple of the world's better surgeons for their hospitals, for example. Most of them go to America. Albania gets competitively outbid.

The sort of "human capital bankruptcy" scenario described happens all the time. The constant outflux of trained and educated and motivated people from developing countries to the developed world is a major part of why those parts of the world remain perpetually "bankrupt".

> This could have the additional benefit of stifling authoritarian governments—because if citizens aren’t treated well, they could just leave.

Borders can be controlled both at the entry and at the exit. Those authoritarian regimes are never going to allow them to easily leave because it would be self-defeating. And "If authoritarian regimes opened their borders they might cease to be authoritarian." is not much of a useful observation.

This story makes the fundamental error of assuming that a country and it's government are the same.