I downgraded my whole Infrastructure back to 6.7.1 and it is now working fine. Fields missing means, that the data was visible through kibana but only the timestamps, no other fields where inserted. I just had the @timestamp field.
I've tried to insert the template manually but this didn't solve my problem. Personally for me it is very hard to get around what ILM is doing. I get the point from the documentation, but I coudn't get the point how ILM is connecting my costum-named-template to an exsisting ILM-Policy? Is it actually doing this? Or just ignoring the ILM-Polcy if I already have inserted the defaut one? How can I change the name of the default policy? So many questions... 