I agree with your overall point that to put the fire out, you've got to extinguish all the embers.
But, as an alternative to synchronization, you could also require vaccinations for everyone who travels.
If a country gets it under control within its borders, it could require visitors to show proof they either (1) have tested positive for the disease and then recovered or (2) have been vaccinated recently enough (but not too recently) and also tested negative for the disease.
That way, you'd know they do not have it and have some degree of protection against picking it up while traveling to your country.
But, as an alternative to synchronization, you could also require vaccinations for everyone who travels.
If a country gets it under control within its borders, it could require visitors to show proof they either (1) have tested positive for the disease and then recovered or (2) have been vaccinated recently enough (but not too recently) and also tested negative for the disease.
That way, you'd know they do not have it and have some degree of protection against picking it up while traveling to your country.