Shit's not gonna work without some form of quests that require team work.
Here is a quest suggestion I've once made somewhere:
1. Find/assemble/repair radio
2. Try to catch the periodic radio broadcast that will give new coordinates and times to designate a meeting location every week. The broadcast could be daily or once every 4/6/8/12 hours
3. Meeting location could be somewhere on one of the islands or any place on the rest of the map. A boat or a helicopter will come to the designated location, wait for a very short while, maybe 10, maybe 30 minutes, and then leave. If anyone shoots at the vehicle, it will attempt to leave immediately while the NPCs in/on it will return fire. You will also need to approach the vehicle unarmed (not simply holstered). If you approach with one equipped, it will open fire at you.
4. Once you "activate" the vehicle, <something happens> that require you to survive until the next meeting next week without getting bitten. Maybe you will be administered an experimental drug. Maybe they are taking a blood sample from you. I don't know, doesn't matter. What matters is the game mechanic.
5. If you make it to the next meeting, you will be given a "science kit" (equipped on the tool belt) and will be required to do, I don't know, something tedious and dangerous. Eg. collect blood samples from 100 zombies or 20 players or something like that. 20 players sounds more challenging and team-worky. You will also be given the time and coordinates for the daily batch of supplies that will be dropped off from a plane for the next seven days. Drop time will vary between 10 am to 16 pm. Supplies will potentially include top gear and medicine that can not be obtained anywhere else. No more item spawn hoarding, or heavily toned down.
6. After you get blood samples from 100 zombies or 20 players, you go to a third meeting for more tedious shit? IDK.
If the game had RPG elements and you could slowly gain low profile abilities, that would be an incentive of its own to survive. Since they are aiming at player-built bases with standalone version, maybe one of the advanced goals could be to build and maintain "safe zones". After that, there could be the occasional player-hunts, for instance. Or there could be multiple radio broadcasts by competing factions that put players up against each other.
Simple things like these would enhance both team work and pvp aspects, imo.