I have the same problem after accidentally deleting the current image file.
Like in the screenshot of smiene07 the button to select another image is no longer displayed.
I tried to restore the Background-Image file from a backup and reinstalled the theme. This didn’t help …
Are these settings saved elsewhere and if so, where?
I’m using a child-theme but Is’s stll the same if I’m switching to the original theme.
I’m also using the plugin WP-Super-Cache and deleted the cache.
Interestingly, the image in the header banner (also deleted ) doesn’t causes this effect: there I was able to simply select a new image via the customizer.
As a “quick&dirty” solution I’ve fixed this via custom ccs but I like to get back this useful feature of the customizer! ??