The problem is that letting the virus continue to circulate unchecked in the developing world just gives it more opportunities to mutate into new, possibly vaccine-resistant variants. Variants can then spread back to infect the first world, and as we've already seen, this can happen quite quickly.
A global vaccination strategy is needed if we really want to get on top of Covid for the long term.
There are two possible strategies for the problem you describe:
Either you vaccinate everyone world-wide at the same time as quickly as possible. Then mutations that make the virus resistant to vaccinations only have the time to occur during the time it takes to vaccinate. However, we do not have sufficient production capacity for doing this. We would have to wait a few years before even starting the vaccinations, otherwise the virus would "mutate away" from the vaccine. Or it would mutate away, if the campaign took too long, because everyone is waiting for the slow drip of doses...
Or you do divide and conquer: Put up travel restrictions, quarantine and testing regimes to split up the problem into smaller areas and subsets of the global population. Quickly vaccinate each of those, so that the virus doesn't have time to mutate away. If possible eradicate the virus in those regions. Then start with the next set of regions and populations, possibly adapting the vaccine to the variants that have occured there. Eradicate the virus there. Then lift the travel restrictions between the regions where the virus has been eradicated. Repeat until done.
Accidentally, the latter one is what we are doing. However, I fear the travel restrictions are too lax to really make it work, because it isn't really planned, just accidential.
A global vaccination strategy is needed if we really want to get on top of Covid for the long term.