As has been pointed out numerous times, zombie apocalypses of the classic zombie-bites-you-and-you-become-a-zombie are stupid and would never work in real life. Biting is a horribly inefficient method for a pathogen to spread, the infected are obvious and scary enough that you could easily lockdown any zone that screwed up badly enough to be taken over, and the undead are no match for a trained army in combat.
Common answers for how to change a zombie virus to let it realistically create an apocalyptic scenario generally revolve around making it stealthier or giving it a long incubation period. However, in those cases there is no real "game": the moment people realize what the virus can do, anyone infected will be treated like a potential zombie. Either infected areas will be quarantined and the threat will end, or, if the latency is long enough for it to infect most humans by the time people realize what it does, humanity has already lost before the war begins.
Besides, that kind of defeats the whole fun of zombies, where there is a conflict between the infected and the uninfected.
What I'm aiming for is a zombie virus that is close enough to classic zombies that they can still be called "zombies", but they can still potentially destroy civilization, not by force, but due to politics - however, if the leadership plays their cards right, the zombies will lose.*
Location should not be a significant factor (i.e. "one country gets infected and others do not, now there is a war between infected and uninfected countries.") This diverges from the point of the story, which is more about the interactions between politicians, demographic groups, and the "viral" transmission of opinions rather than military-type strategy.
Symptom onset should follow a normal distribution. What this means is that, while there can be an "average" incubation period, symptom development can happen at any point - there should not be a long latency where nobody develops a symptom, followed by a sudden spike where many people develop a symptom. Most real-life bacterial or viral illnesses work this way, though they differ tremendously in how long the average onset can take, whether the scale is in terms of days (as in flu or cold viruses), months (as with rabies) or years (as with HIV).
Biting should be a major vector, but it doesn't have to be the only vector.
The virus does not need to affect everyone the same way, and symptom development can be as complex as you like within the above constraints. It does not need to make 100% of infected become mindless and violent. It can have various interactions with genetic factors, lifestyles, drugs, etc. Have fun with this.
However, everybody being infected SHOULD result in the collapse of civilization. This is a bad virus, not a skin condition that is blown out of proportion. (But some denialists might THINK it is.)
(*This is for a scenario in a simulation game, where the player controls a politician during various kinds of crises, with the ultimate objective of maximizing their popularity, so the idea is to make it challenging but not unwinnable. However, the same mechanics could be used in any story about politics and zombies, so it's more a worldbuilding question than a game design one.)