Yeah I’ve never heard of this either. Meghan O’Donoghue just ran a horse at Jersey Fresh with Polyflex, and they certainly didn’t remove those.
No idea about things like endurance, but the only time I know of when Dressage, H/J, or eventing horses may (not always) have their shoes removed for vetting is when they make an olympic or pan am team and are subject to examination by the team vet prior to official appointment from the short/long list to the team (or when they’re sold and need a PPE).