No, the caches API is not called when the site cannot be reached.
It is simply not an automatic cache mechanism.
It is an author-controlled, rather than browser-controlled caching mechanism. The author (the developer) decides what to cache, when to cache (and can version things, too) and what to load from that cache and when.
This is not a Chromium issue (it should happen in every browser that supports the API), this is what the API is intended to do. If the service worker is entirely generated by that static site generator, it might be a configuration issue, or a bug in the generator.