The stigma does need to stay in the flower, but the pollen/stamens can be removed to make it easier to get them to the other plant's stigma. You can try to take pollen from each plant and use it to pollinate the other''s stigma/s (so both plants have a chance at being the 'mom' and the 'dad' with each other). Since some daylilies are better able to be a mom-pod parent or dad-pollen parent, you'd increase your chances by using their pollen both ways with each other. Maybe some of the hybridizers here will also have information about the mom-pod and dad-pollen strength of your particular daylilies.
I've read that it only takes a few days to see a pod form (and in the meantime the dying flower falls off by its own, and should not be removed earlier by hand). But, I also wonder what is the normal range of days to see a pod form, rather than just how long the pod takes to mature and ripen its seeds.
Some say to cover the flower with a fine netting (held in place by tape around it at the stem?) to keep the flowers from being further pollinated by any other pollen that may be nearby.
The more people ask about things, and share their knowledge, the more we all learn, so thanks for asking your questions! Your excitement is contagious
and that inspires us all.