I set up a staging site (but I couldn’t use the staging plugin you recommended without significant hassle as it disabled pretty links and the Community Calendar requires them) and eventually discovered a conflict with the Toolset (WP Types) Layout plugin. Whenever I disabled that, the Community Calendar links worked.
I’ve submitted a support request to Toolset/WP Types (On the Go Systems) as they had issues with the Calendar plugin a couple of years ago and had a workaround. Hopefully, they can do the same for the Community plugin as well.