Curing the meat right away before you take the time to stuff the sausage is a good idea if it will take a long time. Stuffing sausage by hand with your finger and a broken bottle might take all day if you happen to live in the Soviet Union or the like. Getting Cure on the meat as soon as you could may have been a very necessary food safety step when you were making sausage to survive with no special equipment, just a knife and meat. Cutting mince by hand without a fancy meat grinder takes time.
I make sausage by cutting the meat with just a knife and hunk of oak firewood which has been flattened. It takes time, so you either freeze your fingers or throw some cure on the meat check more recipes at
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That's my guess. Putting the cure on early helps me make sausage slowly without worrying so much about the meat staying freezing cold or turning brown.