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.
The unfortunate side effect if this became standard is that there may be far less motivation for wealthier nations to help poorer nations vaccinate everybody.
I don't entirely disagree that it might remove some motivation. But right now the whole world is in crisis, and I don't think gaining leverage by tying everyone's fates together is the best solution.
Anyway, the travel restrictions I mentioned would hardly be bulletproof. Vaccinations don't confer 100% immunity, and tests aren't 100% accurate. So countries could still reduce their risk by supporting worldwide vaccination.
Also, this might be a moot question if we can't produce enough vaccines to do it all at once.
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.