I still kind of swaddle Tyler for his nap, and he'll be 2 this month. I did a full Aussie swaddle with a bedsheet until he was 8 mo, then I very slowly transitioned to wrapping his legs with a normal thin blanket. He didn't need it anymore, but the act of swaddling was a strong sleep cue for him so I kept doing it. Now the blanket is his lovey...after I put him in the crib, he usually pulls it off his legs and cuddles up with it.
If your DD's startle reflex is still waking her, then I don't think it's time to think about weaning the swaddle just yet. As for the daycare, it's their job to do what you ask, even if they think it's weird...she's YOUR child. If you're concerned, you might stop by unannounced during nap time every so often to see how it's going.