> Think about switching your router every 10 minutes.

Cellular and WiFi AP roaming are solved problems. For SpaceX, this will be rather straightforward as the design specifies fixed ground stations that will handle negotiating/maintaining upstream connection(s).

> Or you get a rainy day and your signal clarity goes down.

That's more likely to be an issue, but having multiple satellites overhead, plus the fact that the constellation will be located in LEO (thus higher signal strength due to proximity) will help mitigate signal attenuation due to clouds/precipitation.