Why would a bike in storage deteriorate that much more quickly than one in use? You do need to store the tires inflated, but I use one set of tires for years, and I keep the bike outdoors in all weather and use it daily.

I do get the impression bike tires have gotten better over the years, because I don't remember tires being that robust in my childhood (but, well, memory and all...).

Tyres may be left in a garage or shed, where they're susceptible to direct sunlight and/or ozone.

Ozone? Could you please elaborate? What's with storage that increases ozone exposure?

Electric motors in a garage or storage room (e.g. for a heating system) might be a source of ozone, or sunlight on oils and solvents. The storage itself isn't the problem, it's the higher-than-average level of ozone in typical places rarely used bicycles might be stored.

https://en.wikipedia.org/wiki/Ozone_cracking

Do tires exist that don't have these problems? Perhaps something made of a hard foam that doesn't deteriorate while stored?

Yes they do, for example: http://www.tannus.com

The trade off is that they're somewhat heavier and have more rolling resistance than a good air-filled tire.

I've found that keeping tires out of the sun makes the biggest difference. If kept out of the sun, bike tires can last a long time. One time I bought a bunch at the same time, and over the years I've regularly gotten one or two out of storage and found them satisfactory.

They had hard foam tire bikes when I was a kid but they didn't last long. And solid rubber tires are heavy. Decreasing rotating mass on a bicycle is always a good thing.