The jobs that pay well enough to afford housing elsewhere are in those cities.

Not at all. Median household income in Kansas City is $45,000. The median home price is $192,000 (4x income). Median income in San Francisco is slightly over double at $95,000. But the median house price is more than quadruple, at $1.7 million (18x income). Median household income in Manhattan is $85,000. Median home price is $1,000,000 (11x income). DC is a bit better. Median income is $85,000 and median home price is around $620,000 (7x income). Of course it goes without saying the median house in Kansas City is bigger than the median houses in those cities.

Also, the income difference is less than a factor of two for the same person. 57% of people in San Francisco have a college degree, versus 45% of people in Kansas City. So part of the difference in median income is due to the difference in education, not the difference in local salaries.

I don't think anyone chooses Kansas or Missouri (depending on which Kansas City you are talking about) as a place to move to.

In part because there are an order of magnitude more high paying jobs in first-tier cities compared to cities in Kansas or Missouri.

Knowing this, do most people just move to the Bay for a short period of time?

Americans are moving away from the bay area. Those who stay have immigrated from another country.