This has been a longtime issue with SmartThings: the original architecture was highly cloud-dependent. 
As you note, it definitely doesn’t have to be. Many other systems, including Vera, Apple’s HomeKit, Homeseer and Hubitat run without the Internet after initial setup except for voice commands and some third party integrations. (In fact, Hubitat was founded by former SmartThings users who wanted more local operation.) You may have to use the Internet for some initial setup with these systems, but once a routine is up and running, that will continue even if you lose your internet connection. In the case of HomeKit, even its app still works as long as your phone/tablet is on the same local WiFi as your devices. 
The very good news in this regard is that SmartThings is at this very moment nearing the end of a multi year project to a new architecture which will be much less cloud-dependent. I don’t know about the app, but devices and routines will continue to operate if your internet goes out. Much of it has been in a public developer beta for a few months, so it’s real, if not yet complete: there are community members using and writing about it, and you can try it yourself if you want, although it’s still glitchy.
So the trajectory is good to resolve this, although we don’t have an exact timeline and we don’t know all the details yet, like if the app will still require an active internet connection. Make no assumptions. But at least SmartThings is clearly aware of the issue you ran into and is actively working on improving it.
So you have a choice. Move now to one of the competitors which already has local processing or wait for ST to complete their transition.
Here are some official announcements on the project. The topic title is a clickable link.
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