This is such an obvious (and obviously good) idea, there must be a reason it's not done. Is it because it's difficult to get glass with the right thermal insulation properties?

If you go to any convenience store, you'll see this exact solution keeping drinks cold. The problem is that the refrigeration unit has to be running all the time or heat will enter through the glass.

Heat via radiation? So would blinds not work?

No Door Storage? That could be a very good reason not for see through doors. Replacing them with glass makes it very difficult to see what is on the interior vs what is on the door.

Double- and triple-pane windows are a thing, so I doubt durability or cost is a problem. Are triple-pane windows that much worse than refrigerator doors?