Why does aluminum foil on a pizza stone ruin the pizza?

To understand what's happening here, we should first have a look at what a pizza stone does.

A pizza stone is made from a semi- permeable material with a high thermal capacity, or, plainly put, can store heat and soak up humidity. This means it ensures constant heat at the bottom, plus it buffers the wetness of the dough, giving your pizza a crisp, yet fluffy bottom.

This is why your foil causes the problem. Where the stone can absorb water (and steam), the foil is watertight. Which, as you noticed, can mean a soggy bottom - the humidity from the dough and sauce has nowhere to go on the underside.

Unlike the foil, the parchment is not completely watertight (especially not "steam-tight"), so the stone can still do its job.

If you have trouble with sticky pizza and like easier cleanup, parchment is the way to go. Alternatively, a generous dose of (coarse) flour, semolina or corn flour can help a lot. And don't worry if your stone gets a few stains, there is no need to scrub and clean it every time, just give it a quick wipe once it's dry again. From time to time, you can also burn off residue, if it bothers you.

A press toaster has a different working principle: Its plates are smooth and you are roasting your food only for a comparatively short time. And if you looked closely, you might have noticed a lot of steam either during toasting, but especially when opening it. So the foil has roughly the same properties as the toaster plates, hence you'll get the same result with or without it.

Stephie's point about the stone's capability to disperse humidity from the dough is probably the most important. Additionally, I'd suspect that the foil prevents the stone from getting up to sufficient temperature during pre-heating: aluminium is a good reflector, not only of visible light but also of thermal radiation. So by wrapping the stone in foil, you lose much of the heat transfer from the top of the oven to the stone. The stone has a considerable heat capacity. It will still reach the temperature of the rest of the oven eventually, but only after you've left in in for a long time.

If it's permeability then baking steel will never work. I tried steel plates and they are awesome, with or without foil. There are three types of heat transfer: conduction, convection and radiation. Between stone and dough there isn't enough air so the focus here should be conduction and radiation. The stone itself is not a good heat conductor so it cannot transfer the heat to the dough by conduction very well. With foil between dough and stone, radiation is almost gone. That's why the underside isn't cooked. There is no good heat transfer between stone and the bottom side of dough. Parchment paper doesn't block radiation as much as foil, therefore the dough can be cooked by radiation heat from the stone. Steel is excellent heat conductor compared to stone. Even with foils that block radiation it can still cook the dough by conduction.