Polyvinylacrylate, or whatever you want to call it (acrylic, plexiglass, etc.) is both transparent and fairly tough. I expect it’s probably more expensive, but for the amount of material in a CD case, it can’t cost all that much.
I believe those are made of polypropylene which is more flexible (less brittle). It translucent but not transparent.
Or alternately, try flying a kids rubber band powered plane made out of that solid oak chair
and you will quickly see why the toy is made from the light but breakable balsa
Poly(methyl_methacrylate). I don’t think they are as stiff as polystyrene.
CD are of course made to a price. Years ago when I used to teach commercial courses on Unix system admin on the side (good little earner that one) and software came on CDs, I used to joke with the class about the price of CDs. Amusing reality. Question - what is the most expensive part of a purchased CD? The answer is the printed label. The case comes next, and the CD proper the cheapest bit. What this tells you is that the case is cheaper than the printed paper inside it, and the manufactures clearly worked hard to get it that cheap, whilst retaining the aesthetics of a crystal clear case.
The long-box format seemed to be the preferred packaging in the US in the early days, but never really took off elsewhere. Even then, it had the cheap plastic case inside. The all cardboard boxes and sleeves never really took off either.
I used to keep all the CD cases software came in and take them home to replace the cases I trod on.