Use a sheet of nonstick foil instead. Much better than a screen. Dough can't sink into foil and get stuck. Foil slides right off a peel, slicker than snot on a doorknob.
Bake the pizza for two minutes on the foil. This will firm it up so you can remove the foil. Then pull the foil out if you want direct contact with your stone or steel.
First, I use perforated disks and pans to allow moisture to escape the skin as it's baking for a crispier crust. A solid sheet of aluminum foil defeats that objective like nothing else.
Second, I don't have a problem with dough sinking into the holes. Even when I proofed skins for up to an hour on my old perforated pans, I didn't have that problem.
Third, I'm not using pans with a stone.
My preference for pans never changed. They're immensely more convenient than a stone, and Detroit style and pan pizzas are what I want to produce at least half the time anyway. The only thing that happened when I acquired the SiC shelf was a strengthened resolution to use SiC over other materials when I want to bake in that manner. Its properties are ideal for that purpose, but that purpose is diluted by all the other kinds of pizzas I like to bake.
The reasons for using a pan, perforated or in general, are for convenience and reduced preheating.
That's not to say I'm against using a screen on a stone. For some it makes sense. For example, if you didn't want to constantly dispose of parchment paper, which would be a better option than foil from the standpoint of moisture absorption.
Of course without a stone, highly reflective aluminum foil would make a terrible pizza baking surface.