So it's perfectly reasonable for the furnace to turn off when it is disconnected, because disconnection would be a very strong signal for an error state instead of regular intermittent network/service issues.
You could in theory put one next to the furnace in your machine closet but that would be dumb and expensive
While it is controllable via the cloud, even without wifi it continues to function.
Certainly, the standard smart thermostat set up is that your ecobee is connected to the Internet, but controls the furnace using good old-fashioned signal wires
The overwhelmingly most common connection between a thermostat and furnace is a contact closure when calling for heat, with no ability to differentiate between “thermostat is present but not calling for heat” and “thermostat is not present” as both present as "these T-T contacts are not closed/shorted together".