My parents leave a pair of velcro running shoes at daycare for use in the house. They change into them when they arrive and when they leave. The shoes they came in are put on for outside play. Again my preference is for runningshoes but will allow closed toe sandals.
For non walking infants the robeez slippers are fine but once they are intermixing with the other for activities I want shoes for them - ie about 15-18 months. We run, jump, drop things, get our toes stepped on, stub them, etc. and shoes help protect from this and support the feet for jumping. We have carpet in the basement playroom but hardwood floors up. The shoe policy developed after a child that was 18 months got what was called a toddler stress fracture (broken ankle) from jumping down off the mantle at her grandparents house (just under a foot jump - ie like jumping off the bottom step). She was in bare feet at the time and the doctor said for running and jumping she should have proper running shoes to cushion the blow just as we as adults wear running shoes when we work out. Have had the runningshoes policy in place ever since. If a parent truly balks at it and insists on the robeez that is their choice but the shoe policy is in my contract and an ammendment is written in their contract page in the space reserved for doing that. I am in my sock feet or bare feet afternoons at my computer and sometimes kick them off in the basement but I am not the one building towers of wooden blocks several feet high or jumping from makebelieve lily pad to the next or moving so quickly I miss corners and stub my toe - although I have done my fair share of that for sure. I have bare feet or shoes on when carrying the babies.