Space fountains and launch loops can have a very high throughput if you can get them to work - then the main problem becomes building in space accommodation fast enough.

For one fiction yet science based case of moving most of Earths population off planet during a short time window, see the following Orions Arm article: https://www.orionsarm.com/eg-article/49b46fd2198ed

5.5 bilion people over 20 years.