It would be interesting to find out how effective patching methods are in place with satellites considering their somewhat unique location.
And in particular for Starlink, a constellation providing internet access, network capability is absolutely not an issue from an operator perspective. Even without the laser intersatellite link, the satellites can be contacted a fair portion of their time and have more than enough bandwidth to deal with software upgrades. Once they have laser comms, they will have 24/7 access to them, bar the odd failures.