I think reddit is being naive by charging per API request. Do that, and application developers will try to reduce them - for example, caching popular subreddits or posts (ie. "this is /r/technology as it looked when we last crawled 15 minutes ago").

If Christian added a cache layer on his own server he could easily make the finances work.

But... Thats in nobodies interest - Users end up with stale content, Reddit looses users due to stale content and loses revenue due to Christian extending caching times to save money. Also, Christian will make uncachable requests, like for example voting, hard to do, which again hurts reddit as a platform.