Excuse me Marc. I should not have said most, as there are composites in a few parts of the world. From what I have read Korea, parts of China(not Mongolia) used fish glue. Even the Turks used a hide and fish glue mix. These are 3 popular composite bow types. I am no expert like some of the ATARN folks, just like to dabble.
There is no drawback to letting sinew season for a long time(except boredom), but the sinew will continue to cure for a long time even if you start shooting the bow in a week or two. Look at some of the California bows in museums with a huge amount of reflex that were said to have cured for a week. I have seen it in juniper and incense cedar bows with heavy sinew backings. I started shooting them in about two weeks and over the next year the reflex increases more every time they are left unstrung for a bit.