I do believe that LH has no concept of customer care. They are very arrogant in the extreme and simply refuse to accept, or believe, they can make mistakes. Yet they do, continually.
My wife had seat 4a, booked 3 months in advance. She took her seat, got up to talk to someone she saw and knew, returned to find someone else sitting in it. His boarding pass also had 4a printed on it and he refused to move. The stewardess refused to help, took my wife’s bag and moved to to the back of the C cabin and put her in the middle row. Worse, she refused to giver he back her boarding pass, so no proof.
Spoke to LH customer care (the title is an oxymoron) who said my wife was mistaken, she was seated in row 14. I reserved the seat through Swiss senator service, they even acknowledge it was 4A but LH refuse to accept it and will not admit they made a mistake.
Cannot be bothered to take it further so will just not make the effort to fly LH again.
In your case, press for compensation. There is a website that helps in making claims, and can verify if the delay was down to the airline or really was the weather. Google them and go against LH and enjoy the compensation, which will be in cash.