mybinder.org is a hosted instance (aka a BinderHub) of an open source project called Binder. It so happens that mybinder.org is also hosted by the Binder community. People often use Binder and mybinder.org interchangeably but that’s not correct. There are several other BinderHubs and more are likely to be available. For more information see the binder project FAQ.
You can just use the hub argument in various functions (generate_badge, build_binder) and swap out to another binder hub. If your org has the resources (funds + dev ops people), you can even host your own BinderHub that has more resources than the public BinderHub.
All is not lost. The effort you put into creating a description and Dockerfile will make it easy for people to spin up a Docker instance locally on their machines. It’s not as slick but still better than leaving your setup less documented.
Sometimes the servers are down or overloaded. You can check out the status dashboard. Refreshing the page will usually fix things if the launch doesn’t automatically work.