i guess it could be missing rights on the storage folder.
I have no experiences with Hetzner's webhosting but part of my apps run/ran with success on other german hosters (netcup, all-inkl).
Sometimes it's important to configure PHP not to run as a apache module (as user www-data) when you put your sources in your web-directory with your main Hetzner-account. Take the FPM- or FastCGI-Mode instead.
Having access to SSH, git and composer is also very helpful
My workflow is:
SSH-login to your webhosting folders
getting the sources from the Git repository
composer install
create your db
edit your .env file
php artisan migrate
pointing your (sub-)domain to the public -directory with PHP running in FPM-Mode
this works out of the box for me on my webhosting apps.