Planetside 2 has its own problems, most of its hitdetection is client side...meaning when your computer says you hit something the server accepts it.
That is true! However, first of all you don't have to do clientside hit detection. Especially if the amount of projectiles per time is low, and/or the player entity can't rapidly change direction or speed. Planetside the cards stacked against it there.
Second, even with clientside hit detection you can still do serverside sanity checks. For example, if a client sends hits, you can check that he doesn't exceed the firing rate of his weapon, or that he's even able to have LOS to his target, or that he doesn't hit 20 different targets spread out far apart in short succession.
Many shooters have a prediction-system for movements in order to compensate pings. Its those situations you experience where you thought you shoot first but didnt kill him but pumped all your magazine in him, but you still died first.
Running around the corner and being dead in less then a second and it wasnt a headshot or something like that.
The need for prediction is not tied to any particular networking architecture. Also, what you described is an mostly effect of client side hit detection, not prediction - although prediction can exacerbate the issue. Then there's interpolation which Elite does too, you can see it in action in unhealthy instances, when ships jump around every second. They keep moving between updates based on their previous speed. Essentially, it's prediction, rendered visibly over multiple frames.
So yeah, Elite does that too and it's P2P. Another example: Space Beast Terror Fright, a pure P2P game (and very fun!). They too have to do prediction. They too have the same issues: Low player numbers (though that is ok considering their game premise), and consistency issues. Plus they can't support partition tolerance (i.e. people leaving the game) but that's basically a lack of understanding on the developer's part.
P2P has its own weaknesses but there is no better system. If you want a better Networking...invent lightspeed internet connection, then all our problems regarding that will be gone.
Not so. P2P simply doesn't scale. Even in a LAN you hit bandwidth limits pretty quickly. The amount of bandwidth required raises exponentially with each added peer. And while with C/S the bandwidth-over-players curve is also not exactly linear, it is far, far less steep. And yes, that difference is huge.