Look into integrating (syncing draws to) a club provided Google Calendar which can then be embedded on the club's schedule screen replacing the current homegrown calendar for clubs choosing to integrate.
It may be possible to have the club's google calendar subscribe to the club's All Events calendar instead of needing an API level integration. I'm not sure if the embed would include subbed calendar events though. Will need some testing.
Ideally for every event (league, competition), the administrator can choose to publish it to one or more club google calendars. So on our "Integrations" screen, clubs can setup one or more Google Calendars (potentially non-google calendars in the future?) using API keys and then they'll be able to select which events (in the current season) have their schedule sent to it.