Great, so my problem was to use https://www.uniprot.org/proteomes instead of https://www.uniprot.org/uniprotkb that do not deliver the same data. Sounds weird to me to not be able to download fasta proteome from the proteomes side of uniprot!
I have finally used API URL with the snippet they provide: https://www.uniprot.org/help/api_queries (2.2 Large number of results: use pagination).