In your situation @Achim149 I think if you are a in a strange place, late at night with a little one and unexpectedly need to stay over, I would always stay in a hotel rather than a short term rental.
If you are going to book a STR late at night then definitely call the host before setting off to the listing. If you can't reach them then call Airbnb.
I am sorry you booked with a host that provided such poor service, if he wasn't available then he shouldn't have had his listing live.
It is not at all clear from your post as to what Airbnb advised when you called them to let them know the place you booked was not available. Normally they would help you find an alternative or agree a contribution towards your hotel. Airbnb will give you your money back and may provide an additional small credit towards your hotel.
Make sure you leave an honest review for the host
What did Airbnb advise at the time?