It depends on how the zombies are made - if it’s one of those “everyone who dies always comes back as a zombie” deals, the fighting will never end until the last living person is gone.
Death Stranding had something like this, except people became nukes instead of zombies when they died. Assuming it isn’t an instant switch from death to walking corpse it would probably be handled the same way with corpse disposal teams transporting bodies to an incinerator ASAP.
I’ve never seen / heard / read any zombie fiction where they were completely unkillable. The standard zombie fiction has them gone for good if you kill the brain. Sure, everybody who dies comes back as a zombie, but that just means you kill the first wave over a few years, and then make sure that any time anybody dies their brains are perforated and then they’re cremated.
There’s decent evidence for how humans would handle that situation. Ebola used to be a real problem in Sudan / Congo. Part of the problem was that typical funeral rites involved washing the dead bodies by hand. That spread the disease and more people died. Once people realized that they couldn’t do that without spreading the disease, they adapted. At a certain point the survivors would just have standard death practices that ensured that nobody who died came back as a zombie.
There are some fictional villains that are unkillable. Some that can even eventually self-assemble if you do something like cremate or atomize them. But, they’re individual villains. I’ve never heard of anything like that for hordes of zombies.
Besides, even if zombies were completely unkillable, they’re dumb. Herd them into a mine and then seal it. There are mines that are currently (or were recently) used to store Helium. If they’re so enclosed that not even the second smallest element can escape, they’re going to keep Zombies enclosed too.
I don’t believe that if everyone who died came back as a zombie (and no, zombies don’t come back - once you kill them they’re gone) that the entire world would be coordinated enough to keep them from ever becoming a threat - certain countries might be diligent enough to make sure most corpses had the head destroyed immediately, but countries with less resources would become overrun, just like disease hits them harder now. But you’re right, they could probably be kept in check by some countries indefinitely.
I think countries with fewer resources would be better off. They’re not as interconnected and dependent as the richer countries. Plus, people there are more used to hardship and probably more likely to have encountered death, even if it’s just an animal.
People in those countries are used to having to get their water from a well instead of just turning on a tap. They’re used to electricity being unreliable or going out. They’re used to not getting around by car. They don’t rely on supermarkets with their just-in-time supply chains delivering goods coming from other countries to get fed.
If there was a mild zombie outbreak, a more developed country might handle it better because they could mobilize armed forces with body armor, guns and lots of bullets. They’d have great communication infrastructure to coordinate their response, and so on.
But, if it was a devastating attack where half the population or more was dead, it would be so much worse. People in the developed world rely on modern conveniences and have never had to do without: tap water, well stocked grocery stores, reliable Internet access, reliable electricity, gas stations always having gas, etc. If the power plants started failing because too many who knew how to operate them had been zombified, that would have knock-on effects to everything else. We saw just how disruptive COVID was to supply chains, and that was a plan that a committee thought out and implemented, trying to think about all those difficulties.
