After playing with the TYPO3 log settings I had the following fatal error in the log:
nginx stdout | 2017/02/01 09:29:08 [error] 15#15: *840 FastCGI sent in stderr: "PHP message: PHP Fatal error: Uncaught TYPO3\CMS\Core\Error\Exception: PHP Warning: constant(): Couldn't find constant LOG_ in /var/www/site/htdocs/typo3_src/typo3/sysext/core/Classes/Utility/GeneralUtility.php line 5097 in /var/www/site/htdocs/typo3_src/typo3/sysext/core/Classes/Error/ErrorHandler.php:111
The facility was missing, TYPO3 interpreted it as empty string and thus used the constant LOG_ which does not exist.
After setting it to syslog,LOCAL0 it worked fine.
But in the end I switched to error_log because I had that already configured to be sent to docker's stdout, which in turn is sent to our logging server.