Calibre Connector is up on the app store where it's been for a few weeks. It's been through a couple of iterations with the original January version being rewritten using the SwiftUI framework rather than the UIKit framework, then a couple of bug fixes and enhancements. If you downloaded the early version, the current version is likely more useful.
The app is free and is basically a hobby project to provide much of the functionality provided by the original excellent Calibre Companion. I have a somewhat different spin than the standard Calibre UI and I set it up to automatically download all missing books when syncing rather than manually pick them since that's the way I prefer things to work.
The documentation is at
https://www.phillipwwalker.net/
The link to the opensource project that I created for it is at the bottom of the under the hood page. Anyone who wants to download the source and use it as a starting point for their own projects is more than welcome to.