As the BL mentions (and the T-S article does not, I think), "there appears to be a greater preponderance of 16th century items" in this collection. That is because it is not really a Genizah collection, but rather a trove of fragments buried in the Cairo Jewish cemetery, as described here and here.
Mosseri was a Jewish businessman in Cairo, involved in the Zionist movement there, and he left with his family for Paris in the 1930s. His widow apparently had little interest in the collection, and the process of microfilming it for the IMHM in the 1980s was a fraught and difficult one, as described in the introduction to the (very concise) catalogue of the collection, compiled at the IMHM (Jerusalem 1990).
I spent much of last week cataloguing ketubot from Jewish weddings performed in the Moroccan cities of Rabat and Sale. It was an informative experience for me - I didn't realize there was an organized Jewish community there, performing weddings, with official stamps and all, in 1959.