A week into the evacuation from Kabul, the US military was forced to take a drastic step: stop all flights from Hamid Karzai International Airport for seven hours because there was nowhere for the evacuees to go.

For months, military officials had urged the US State Department to convince other countries to take Afghans at risk from Taliban retaliation. They had largely failed to secure agreements with other countries, prompting officials across the US government to rush to try to...