Hey Lars,
Thanks for reaching out.
We offer this functionality in a premium add-on to The Events Calendar called “Community Events”. You can learn more about Community Events here: https://theeventscalendar.com/product/wordpress-community-events/
↑ You do not need Events Calendar Pro to use this! You just need the free core plugin, “The Events Calendar”.
Community Events lets folks submit events, and yes, you can manually approve submissions before they get posted to the calendar and such.
Please let me know if there are any other issues or questions I can try to help with.
Cheers,
George