I think most of what I have to say here has already been said, but perhaps a slightly different cant to it ... (?)

I generally use a polycro/window insulation ground cloth because on longer trips I'm just pitching my tent --- more. More dirt on the tent bottom. More chance of abrasive ground surfaces, though I do find that rare. But also more chance that I might find an opportunity to cowboy camp, in which case I want the polycro to throw my bag onto and keep my stuff out of the dirt.

One of the last comments was about the ground cloth not extending beyond the tent fly/tarp, and that's a definite, especially if your bathtub floor is no longer 100% waterproof. Since I use my ground cloth without the tent sometimes, I don't have it cut exactly to the size of my tent, so have to remember to tuck it in on rainy evenings. Probably not the best, but it works for me.

I find tyvek too heavy and bulky, though as it ages --- depending on the tyvek --- it can certainly get more flexible. I guess the flip side is that with something like polycro, the virtue of it being lightweight/low-mass is that you just don't expect any puncture resistance from it. It's tough stuff, carefully used I've had it last for months on trail, but the way it fails is you get a longish slice in it that grows, or sometimes several of those.

I think the folks that bring no footprint have a great deal of logic on their side; I just have old tents (not so sure how waterproof the floor is anymore), and I prefer to get my polycro muddy rather than my tent floor. And again, cowboy camping.

Hmm, and sometimes the ground cloth is handy when first pitching the tent. My eye will apparently never be calibrated enough to angle the tent to make it maximally level at a site, I always seem to have to lay down first and adjust it. Easier to do this with a ground cloth (always involving four rocks or pieces of gear with the polycro as even slight wind will ripple and move it).

Ah, and it's also an item I can dry out separately. In some conditions the tent floor can be quite wet in the morning, while the tent body is dried out from the breeze. In these conditions I can put my tent away fairly dry with little fuss, and shake/wring much of the water out of the polycro --- and dry that as an individual item at my leisure if I'm inclined to. So I guess there are multiple small benefits from this light piece of gear.