SmartThings is primarily a cloud based system, so it’s not going to meet your first requirement of operating when the internet is unavailable. For example, the Sonoff devices require custom code and all custom code in a SmartThings setup runs in the cloud. So if the Internet was down you would not be able to use the Ikea buttons to control the Sonoff lights.
Also, the SmartThings app requires the internet to control any of your devices even if it is on the same local WiFi. (They didn’t have to design it that way, but they did.)
So I don’t think your wife is going to be happy.
I suggest you return the SmartThings hub.
Since budget is your primary consideration, both Sonoff and Ikea are good candidates, but the trick is what will work together.
If you’re willing to do some programming, you can probably put together the least expensive most reliable Local system using the free Home Assistant software. You’ll be able to use either the Ikea hub with it or a Deconz stick. You will also need a Raspberry Pi as the main server. They have a very active community on Reddit who can help you.
If you want something less technical, I would look into an all Xioami system using their hub. This uses a proprietary zigbee. It’s mostly intended for the Chinese consumer market. Very inexpensive, lots of devices, doesn’t require you to be technical. The problem is their button and app won’t work with Sonoff and they don’t have a comparable inline relay. So it might not be what you’re looking for.
Sonoff itself will require some internet cloud unless you are willing to get technical again, and then you can flash the firmware and use Home assistant or MQTT.
The Reddit Home Automation subforum covers all brands, and might be a good place for you to ask more questions about potential local systems.