Electronic – How does a capacitive multi-touch screen capture multiple triggers

I haven't actually done this, but it seems the problem is the objects you are using are too small and don't have enough ambient capacitance. A human touching something adds capacitive coupling to the environment. Think of the size and surface area difference between a carrot and a carrot+human.

You should be able to use something conductive that is covered by a thin insulating layer, then connect the conducive part to a conductive plate under the iPhone or to ground. In this case "conductive" only needs to be not a good insulator. As you found, even something like a carrot is conductive enough. Try connecting a ground clip to the other end of the carrot, or connect it to the chassis of your machine.

It's hard for me to imagine that your 1 big sheet of metal would work as a touch pad for 3 main reasons

1. The (x, y) extraction from the measurement. Having only one big capacitor won't be accurate enough, and that's why touch panel on the maket place have a grid of capacitors. This is the first problem you are trying to solve now.
2. The touch tracking. It's a fundamental need to detect the position of a touch, but not enough. To build a touch pad you'll also need to follow the touch movement, manage the case of a possible "not desired" second touch, handle the "tip switch" in other words when the touch is leaving the touch pad... and many others.
3. The touch driver. To make your customized touch pad as pointing device, you'll need your OS to understand data from your touch pad. You'll build you own protocol, or used a USB standard for example, which requires a huge work IMHO.

Although, I congratulate you for your effort, and I'm just adding one remark

Designs for capacitive touch screens typically involve a "grid" of sensors that give x y values, but if I am not mistaken, this is only needed to register multiple touches.

No, not really. In an other post that have nothing to deal with yours, I've added 2 pictures that a touch screen controller (with image processros inside) is using as primary data. You would notice how a grid is used to capture the touched area. Imagine a big big finger touching the screen.

It would be seen on many "touch pixels", then an algorithm like a centroid detection, will compute the center (x, y) of the touch area. That's not only for multi-touch.