They are safe - mostly. Most birds never have the least problem. Still, āmostā is not āallā. . Back in the day, several decades ago, they were pretty much mandatory to prove your bird was not wild caught. Basically the solid ring proved the bird was banded right after it hatched. You would NEVER buy a bird that was unbanded - it was considered unethical by the vast majority of people - and at the time it was considered the tip of the sword against illegal and harmful trade in endangered species. Of course taking one off AFTER you buy a bird is a separate issue. Up to you.
Nowadays these have largely been replaced by chips, and there is a bit more of of an international legal structure in place to protect wild birds. My parrots born decades ago had some basic info stamped on the band, like month and year of hatching. Sometimes the breeder would also put their ID on there too. Now the chips have birth date, but also sex, name, home address and phone number. Also anything you care to add. You can buy a device to edit the info, but most people now have their vet do it for them.
My new baby 4 month old African grey came straight from the breeder with a chip, but no permanent band. But I have been following him since two weeks after he hatched. He still has a removable plastic band (kind of like a key ring) that was put on him to ID him after his genetic sexing while still being hand fed with his siblings. But it needs basically 4 hands to take off - 2 each for bird and band. Iām going to have to convince someone to help me!