Webform does not submit when a file field is added
I discovered the issue: I had added a script to my theme to prevent double form submissions: a simple "if the submit button is clicked, disable submit buttons in the form" jquery script.

This script prevented the submit button of the field field from being activated at the same time the main submit button was pressed. This confused Drupal to a sufficient degree that nothing worked anymore.

I fixed it by wrapping the disabling of submit buttons in a window.setTimeout(function yada yada ... , 50).