Wait. Rewind that a bit. No, no, past there:
Okay, hold that thought. Now: how do you kill a zombie?
Answers vary - fire, various religious methods, cut off its head, et cetera. Most agree that the last works, though, or at the very least that caving in the brain will do it. No more brain equals no more zombie, just a lump of now-harmless flesh. (Yeah, don’t eat it, don’t go near it, but it won’t jump up and eat you anymore.)
Okay, now back up to that thought you’re holding. Zombies are rotting. All of them - including that very vital brain. Shambling around will most likely not slow down this process, and many forms of zombie apocalypse don’t re-animate the normally dead, which means that they won’t be embalmed. Sure, some variants of the virus that cause zombies to exist will also slow down decomposition, but they can only do so by so much. After all, they aren’t very good zombies if they aren’t rotting.
Say the embalmed come back to semi-life. Say the virus slows decomposition. Say the infection takes a while to spread around the globe. I’d still say that within ten years at the absolute outset - with a little luck, maybe within less than one - every last zombie will have rotted to a pile of bones.
Survivors, the world is yours. And given that your apocalypse was zombies rather than bombs or war, then there’s probably a fair bit left - buildings, roads, cars, et cetera. Is society as we knew it going to be immediately restored? If we got the the apocalypse stage, then eh, probably not. Is the human race as a whole fucked? It depends on the numbers of survivors - I don’t know enough about genetics to say how many you’d need - but I’d put my money on a bounce-back, there. Even if not, said survivors might just have a decent while left to live and find each other, might even be happy. And the rest of our planet’s flora and fauna, well, according to the rules of most zombie viruses then they’re fine. (They kind of have to be - imagine if you had to worry about zombie bugs. Or even mice. You ever try to keep mice out of a
place when they really want to be in there?) Who knows, the bears or the dolphins might develop sentience.
I still don’t want to look out my window and see zombies, but I don’t think it would be the sign that it’s time for the species to throw in the towel.