Google Calendar sync works only when Google Calendar app is uninstalled

I couldn't get Google Calendar to sync on my Galaxy S3 (with web version of Google Calendar). I did all things recommended on this help page but it didn't help. I also tried removing Google account from the phone and adding it again.

However, during one of the reinstalls (as recommended in Google help page) I suddenly got a notification about an event in my calendar that was created in the web version. The sync worked perfectly fine when the Google Calendar App was removed from my phone. When "Google Calendar App" was uninstalled the pre-installed "S Planner" showed me the notification because the sync kicked in.

I rechecked it both ways from Android app to the web version and vice versa.

I prefer Google Calendar app to "S Planner" but this issue renders Google Calendar app unusable.

I also sent feedback message from the app.

Edit to add: it worked fine until one of the updates of Google Calendar.

Update: user RossC suggested to disable S Planner and test sync, but unfortunately I can't disable it (see screenshot). Disabling related applications (with words calendar, samsung and sync in the title) did not help.

full screenshot