Thanks again for the hint that in principle the pipeline/statefiles should work without my intervention (as long as the h5-file for a particular run lies in the same directory as the unchanged statefile and has a particular (unchanged) name, of course).
I found that I have been writing the statefile on ParaView 5.5.2 Qt5 on my institute and was trying to load that same state on ParaView 5.6 on my private laptop at home. Installing 5.5.2 and loading the state based on data from different runs suddenly worked!
This is great indeed because I was really afraid I couldn’t do this sort of job with ParaView. And without the really handy possibility to convert the many vtu-output files into one h5 in ParaView that probably would not even have worked without a lot of manual modification of the statefile to adjust to each particular run.