It was kind of a scam for which both Airbnb and the High Street Hostel should be credited.
First, the French translation of the room and hostel description only mentioned that the children cannot check-in, but not that they can't stay at the hostel - contrary to the original English text which was sent to me yesterday by Airbnb and which indeed mentioned both interdictions concerning the children.
Second, the hostel informed me only less than 48 hours before arriving at the hostel that the children cannot stay at the hostel whereas I sent to them one month ago an email where I informed them that I will come with my young children. I got no response but about 36 hours before the check-in date informing me that I could not be refunded and I had to find an accommodation at the last moment for the night with my children. They also dishonestly and pathetically tried to convince me that checking in is the same as staying for every single customer, whereas usually in every hostel everywhere on Earth, one person of a family checks in for all the other members of the family.
The hostel not responding to my email, and Airbnb not translating all the children restrictions in the French page: both of them indeed participated in scamming me. Each one knew that the other was wrong, but they agreed to crunch me, to scam me as a customer, not to recognize their faults, not to spend a single penny. They both refused to find me another accommodation (imagine how stressed I was with my kids when my accommodation was cancelled at the last moment despite my one month-old unanswered email). Airbnb did not refund me but they urgently modified the French description of the High Street Hostel by adding the restriction that prevents children from staying at the hostel after I complained to them and asked for being refunded. if they added the restriction after my complaint (I have recorded evidences of before/after content), then it proves that it was not mentioned before!
How can you trust such robbers and liars? if you dislike being scammed and disdained, go on your way!