@ dosjockey, Great post! I will look further along the lines you suggest.
The biggest thing to remember beyond that is that tires are part of your suspension system, and given most haven't used too many different vehicles or set up one vehicle in enough ways, this is where asking opinions comes in handy. On this vehicle, for example, you might find more stability with a little less rigidity, or a little more. It depends on how it's set up; but you're after a very specific adjustment; not some mythical "best" tire.
Unfortunately, I can't give an actual opinion on what to buy for a Mk 7.5 GTI, as I've only had one for a little over three weeks. I have a hypothesis in regard to what it needs, but until it's a theory, it's useless for advice.
You know what you want in a tire; the feel of improved steering precision, so you're closer than most tire buyers ever get to buying the right product for their use case. In order to produce that feel, you'll want to look at points where tires deflect; and the two most common places are the tread section and the bead filler area. This is where manufacturers sacrifice precision for comfort, but that doesn't mean you have to give up comfort if you know what you want.
Understanding how all that data goes together will help you avoid purchase mistakes, and land you with something you like. Try to imagine the forces acting on that tire in your head before and after driving around a bit, and you'll sneak closer to being able to determine which aspect of the carcass or tread is letting you down.