I agree with you and with Dave H that there is much to like about using parchment paper. However, I also believe that there are some drawbacks to using parchment paper under certain conditions, as follows:
1) Most rolls of parchment paper that I have used and researched cannot handle pizza sizes above about 15". To make a pizza above 15", it will usually be necessary to "join" two sheets of parchment paper to handle the larger size. This will usually require some trimming of the joined sheets to the desired size. Most individual sheets of parchment paper (as opposed to rolls) that I have seen advertised at King Arthur and elsewhere are also usually too small or of the wrong dimensions to use to bake pizzas above 15".
2) Many--maybe even most--parchment papers are oven safe to around 400 degrees F. That may limit their application to oven temperatures below 450-500 degrees F. Also, if one is not careful, it is possible for parchment paper to catch on fire. That once happened to me when a sheet of parchment paper accidentally fell through an oven rack as I was removing a pizza from the oven. The parchment paper hit the lower heating coil and burst into flames, setting off my smoke detector. The risk of a sheet of parchment paper catching fire, or burning or blackening at the edges, can be reduced by trimming the parchment paper to a bit larger than the pizza to be baked on it.
3) Parchment paper is expensive on a per-use basis compared with using a peel with a release agent. I have read that some people have found ways to reuse parchment paper, so to the extent that that can be done, the per-use cost will of course go down. But, even then, it will still be a fairly expensive use. Large width parchment paper, to the extent found, is also likely to be more expensive than smaller width parchment paper.
My use of parchment paper is pretty much limited to modest pizza sizes (14" and below) and very high hydration doughs (usually above 65-70%) that I fear will stick to my peel no matter what release agent I use or how careful I might be. This problem will usually be aggravated if there are a lot of toppings to be put on the pizza. Usually, I put some semolina flour on my peel (for its ball bearing effect), put the sheet of parchment paper on the peel, dress the skin on the parchment paper (which I can do pretty much at my leisure and without rushing), trim the sheet of parchment paper to a size a bit larger than the pizza, and slide the pizza on the parchment paper into the oven.