Electronic – How do capacitive screens work with rubber head stylus

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.

It just needs to be thin. The capacitors on a capacitive touch screen appear as a series of very thin (so thin you can see through them) metal contacts on the non-touched side of the screen. The change in capacitance developed by your finger in proximity to these is what it senses, so as long as the material is thin enough to not interfere with this, it should be fine. This is the principle by which screen protectors work.

It should be noted that most capacitive touch screens require a minimum finger area (to reduce false positives) which is produced by the action of squishing your finger against the screen.

I would suggest some sort of thin plastic sheet (less than 1mm thick) with an adhesive backing so that it gets really close to the screen, although in my experience the glass is easier to clean.