As stated before Tesla updates are fully controlled by Tesla.
You can get a notification that an update is available to your car while on the road without any connection to wifi. Some updates when made available are notified to the car through LTE connection but require and request wifi to download (I suspect those are considered critical updates or part of a recall).
Other updates (possibly not as critical ones) appear to check in when the car is connected to wifi. With that it may need a good wifi and at some update levels the car wifi works better than at other update levels, so sometimes changing the connections between SSID-s resets the connection and may trigger an update check, other times rebooting the car may trigger it, but no matter what you do, if Tesla has not made one available for your car, it will not appear for you. I do not believe tesla is monitoring your car cameras to see if you do the right update dance around the car to push it for you, so all activities around car are unlikely to have any effect on the availability of updates.
Also important note is that the button between standard and advanced updates is a placebo button, not connected to anything and does not affect any aspect of updates.