I don't think there are any usga rules for shoes. Seen too many pga folks take one, or both shoes/socks off for shots in/near water hazards
I do know of a few local area courses that don't allow bare feet, or spikeless sandals. Sandals are ok if golf specific with spikes.
It's a health/safety issue for them, due to the various chemicals used to treat the turf
My brother and I played at Foothills in CO for the first time on Wednesday and it took just over 5 hours to play the round because they let the group in front of us go out with a group of 5. No ranger, anywhere on the course either. If you are going to play with 5, you should be cognitive of your pace, not them they took their sweet time every hole.
It was a total joke, you can't get into any type of rhythm when it is 5 to 10 minutes between every shot. Seriously doubt I will ever play there again.