Have you tried with different browsers / without browser addons that might impact the sites behavior? Otherwise I don’t know either

I had a broken form on a different website and I used the inspector tool to check the html code. What I didn't know for a long time is that you can actually change the source code and still use the website, so after a lot of trial and error I found one field that was named wrongly, changed the name and was able to send the form.

Although I doubt that the ELSTER form itself is broken, but it's worth a shot.