After some more debugging, I have new information which frankly makes this even more frustrating.
After upgrading to 3.0.210 I also added a new "Repeater" field and added it to a template. Now all the fields after this Repeater cannot be saved, still no error message. I moved the repeater field to the last position in the template and now everything can be saved. I'm still mostly looking for a way to debug this properly.
I also increased the values for max_input_vars but that does not have any effect.