Alexa's hardware is only good enough to recognize its wake word and capture a minute's worth of audio, which it sends to Amazon after it hears the wake word.
None of the processing occurs inside the Echo. It's a thin client.
Even my SmartThings hub, while it is smart, the app on my phone authenticates to Samsung.com first. Then my command is relayed to my hub. my phone can either do wifi or cell for data. So if I unplug the router, the SmartThing can't see samsung.com, and if my phone is on cellular data, it can't see the SmartThing on wifi after it authenticates to Samsung.
So no internet, is a huge bummer.
In my house, I use z-wave light switches. So physically, I still have traditional light switches, even if the hub, wifi, or internet dies. I'd just go back to the old ways.