Been using something similar to this for years. The ties go on it and cannot slip off, and you can fold them back so they don't take up space. Just don't yank them off.
Interesting. Each spoke on my rack angles upwards at the end, which seems to do the trick for keeping them in place. I've had the rack for probably a decade and I've never had a tie fall off.
This might be the right rack for you.
Why don't you get something with hooks?
https://www.amazon.com/Ohuhu-Wooden-Hangers-Rotating-Organizer/dp/B01KH0RJN8