Generally with devices that are WiFi enabled but do not inherently need WiFi to do their main job there are a few possibilities.
• You do not have to configure WiFi to just use the device. It can do all the functions that type of device generally does, with a reasonable interface. You can configure WiFi to gain access to extra features, such as sending notifications or a friendlier interface.
• You have to configure WiFi to initially set up the device and/or to change some settings later, but it does not need to use WiFi during normal operation. WiFi can be used during normal operation for extra features.
• It needs WiFi during normal operation. Without WiFi you can only do at most a basic subset of normal operations.
For devices that do need WiFi it is worth distinguishing between devices that need to use the internet (directly or via an app on the local network) to function. There are devices that need WiFi because they use that to talk to a required app, but neither the device nor the app need internet.
This can be important when you have an internet outage. A device that requires WiFi but does not require internet can often continue to work when your internet is down as long as the outage is upstream from your home router.
For people whose reason for not wanting to configure WiFi is that they don't want the device sending their personal information to the device maker, it is not necessarily enough to just leave omit configuring WiFi. Allegedly some devices will look for open WiFi networks and use those if you don't explicitly configure a WiFi network.
As far as LG washing machines go I believe they fall under the first bullet item above.