this topic shows a solution for the broken POST requests due to Magento 2.3 upgrade. However the solution is breaking the compatibility of my module with Magento <2.3 shops.
Can anyone think of a workaround to support older versions as well?
This solves the problem, but after that your module will not work on Magento 2.2 and earlier. https://magento.stackexchange.com/a/255082/75860
After a quite a bit of research I did not find a solution not breaking the backwards compatibility. I settled with a fork of the original module which I patched as described here.
There is a way to make the fix compatible with Magento 2.2 by getting rid of the '?' in method overrides. However there is no way to make it compatible with Magento 2.0 or 2.1 since the required PHP Versions do not support the method headers you have to match in order to implement the CsrfAwareActionInterface.
The only solution might be something not requiring to implement the CsrfAwareActionInterface at all.
I have updated the original answer to Magento 2.3 upgrade breaks HTTP POST requests to custom module endpoint to include a backwards compatible solution.