Sonoff devices usually do not work as you would like.
Many functions are cloud-dependent, meaning they stop working without internet access.
There are two issues to distinguish… no internet connection and no wifi connection to your LAN for the sonoff device.
If there is no Internet, some functions, depending on the device model, may not work, and scenes will not work either.
LAN mode is intended for communication between devices within the LAN network, but they still must be connected to WiFi and have communication within the LAN network. If your situational model is to turn off the Wi-Fi router and the devices lose connection to the network, it is obvious that they will not work, LAN mode has nothing to do with it.
Many Sonoff devices in 2023 have LAN mode… what does this mean, it means that when the devices are connected to the WiFi router and communication in your LAN network is available, you can then run the ewelink app and still control devices that support LAN mode. But remember that control does not mean support for scenes or other cloud-dependent functions.
The device that replaces the cloud or creates a local cloud is the iHost, which is intended to provide cloud functionality but locally in the LAN without the need for Internet access.
Another example is Sonoff Zigbee Bridge Pro which supports local scenes but only for devices paired to it, i.e. ZigBee.
Many people replace the ewelink cloud with their own automation systems such as Home Assistant in order to, among other things, be able to function when there is no Internet access.
Can the device have a running schedule in LAN mode? Yes, provided that it was set when there was Internet access. Tested on S26R2 and Micro. You cannot edit or add new variables in LAN mode.
Will the schedule work when the device is not only without the Internet but also not connected to Wi-Fi? Yes, tested on S26R2.
But there is a very important thing to note! The device cannot perform a power loss cycle! Otherwise it won’t work.
When the device is connected to Wi-Fi and has access to the cloud, we set the schedule, then turn off Wi-Fi, the device is still able to execute the schedule (tested only a few minutes after going offline), but if there is a power outage, the device will not work. Probably the software not loaded into memory and the boot sequence which will not do anything more than try to connect to Wi-Fi.
The above conclusions are based on short testing based on only two devices and may not be reliable on a large scale and over the long term.