Well I can't think of the qualities of a perfect zombie game, but I can post an idea I had for one that I think is cool.
You are an accomplished con artist, burglar, and all around thief who moves to a relatively small town to wait out the heat from your latest caper. You get plastic surgery done (player chooses new appearance and gender) and a new name (choose from a list of pseudonyms so the voice acting can say your name) and begin another new life in suburbia. You're very rich, so no job necessary, and you pay for everything in cash. In the intro, you carefully and quietly buy all sorts of nice things for your new home, and then go to bed happily that night, ready to enjoy your hedonistic vacation.
You have a horrible nightmare where you're running for your life from crowds of screaming, reaching people, you see cars crash and explode, buildings go up in flames. You're running around the town you just moved into. Then a massive dagger falls from the sky and impales itself in the ground. The dagger is richly ornamented with gold and jewels, but looks ancient and fragile. You wake up in a cold sweat in the afternoon. Hoping to shake off the nightmare, you drive to the store to get some groceries. Town is strangely deserted. Everything is closed. No one is in any of the shops. You wonder if it's a holiday, and struggle to remember which one it could be. You find the grocery store, it's locked up, even though the hours say it should be open.
But then someone comes to the door. He looks... normal, but he stares in a strange way, and he's tearing into and eating a raw steak. You back away, creeped out, as he puts his whole body against the door and chews with his mouth open, staring at you. As you walk to your car, he suddenly shoves the door open, setting off an alarm, and runs after you. You run at breakneck speed, get into your car, and drive away as he still tries to give chase. Very, very freaked out, you drive around to lose the weirdo, then drive home and hide inside. Then that's when you check the news.
The town was put under quarantine since that morning when a trucker tried to deliver to a shop in town. He ran into some locals behaving in strange, deranged ways, and booked it out of town to call the sheriff. The sheriff didn't answer, so he went to the next town over and talked to their police. After another trip to town, an investigation, and running from the locals, the police called for a quarantine of the area and blocked all the roads. Now YOU have a decision to make. Do you escape immediately? Or give in to your thieving ways and start looting the town to make up for what you spent on the house, car, food, and so on? The answer is: loot the town, because simply turning yourself in blows your cover, gets you arrested, and a game over. But for laughs, the option is left there for you to try at least once.
You venture out only at night to avoid being spotted by helicopters. You venture out to find food, water, and loot valuables. It's a stealth game really, but there are "kill teh zombies" moments available in certain buildings where there's a lot of room and not alot of windows. The difficulty curve comes from how careful you are. If you drive around town blaring your horn, zombies will start smashing doors and windows to get at ya. If you set off alarms when looting shops (there's ways to turn off security, and some places have silent alarms) zombies get out and get more active. Zombies forage for food after a while too, so some will simply find there way outside after a few days. And of course, a helicopter will show up in day three with a megaphone and someone yelling for survivors. It doesn't help, naturally. Your character (or the news) doesn't refer to them as zombies until day four or five either, when they start to reach visible levels of decomposition (this makes more sense to me than calling them zombies right away). And they smell bad, your character will complain about it. Infection becomes a problem also, when pieces of zombies start falling off in not so convenient places, and can make your character slip or even sick or to die of infection if he steps onto one of these spots while injured. Carrion animals start hanging around also, attracted by the smell. After a week (or two?) the zombies begin to resemble skeletons, and get harder to kill because most of the brain has rotted away, leaving only instinct and nerves to keep them going.
The "usual" weapons are used, but silver tipped or sterling silver weapons kill zombies very, very easily, which is something for the player to discover on their own. Eventually, your character begins to question what he's doing. Sure he's managed to acquire a lot of wealth, but where is he going to pawn it all? How is he going to get out? Can he get out? Why is he not a zombie? Is he immune, or worse, a carrier for the disease who may not get symptoms until he escapes? At that point, he begins researching the town through satellite internet, and finds out something interesting. Apparently, an artifact was sent to a lab in town for study not too long before the zombie outbreak. It's the dagger from his nightmare, but of course, hand size. The dagger was used in an ancient death cult for sacrifices and tablets associated with it claim it could summon zombies and imbue immortality. There's even a tablet with a story about how the dagger reverted a village of living people into zombies except for the shaman, who had to sacrifice himself with the dagger to restore his people to life. However, people who are killed as zombies cannot come back. Cue possible grief and remorse.
Upon finding that out, the player makes a choice: escape through the sewers with enough valuables to make up for the loss, plus extra. Or, sacrifice yourself to save the remaining people. Or, log onto the internet, find the modern cult associated with the dagger, and convince them to send a few devoted followers in a single engine plane to sacrifice one of themselves to save the town, as long as they can ensure you aren't caught by the police.
Escape ala sewers is the most challenging stealth portion. The player needs to cause a big disturbance above ground to draw the military and police's attention, then attempt to sneak through the dark, slippery, infectious, zombie-infested-and-well-patrolled-by-the-army sewers. Reach a manhole cover on the outskirts of town, and you're home free. Character is in a cut scene later in a diner, with a new randomized face and gender, watching the news about the town with a faint smile. Queue sequel.
Self sacrifice yields some nice benefits. The character wakes up shortly after doing seppuku with the dagger, internal organs all over the floor, immediately vomits a body's worth of blood plus any organs he missed (even his heart and brain...!), utterly freaks out to the edge of insanity for a little while... then starts to realize he isn't dead yet, and if anything, feels 1000% better than ever. Leaves the dagger behind. Still needs to escape through the sewers, but it's harder to get to the sewers because the army is swarming the town to get people out of the town and into quarantine camps. The sewer is somewhat safer, since the army is focused on finding people and getting them out, but the player still has to be careful. Character escapes, has another face change, but also becomes aware that he or she is immortal now. Queue sequel.
Sacrificing someone else results in the above happening to the sacrificee. Now immortal cult member helps the player escape by helping him or her cause a big disturbance (a fire, explosions, who knows) elsewhere in the town so they can sneak into a vehicle waiting for them in the woods. The cult member keeps the dagger, and looks at it with an evil grin as they drive off. Queue sequel.
Sequel would begin by asking the player which decision they took, and would adjust game content to match.