Oooh, various facets to this.
Leg bands have two main types in the US - open and closed. Open bands do run a bit of a risk of catching on something. Closed bands run the risk of, if a bird leg swells for any reason, getting stuck and making the situation worse. (Some people have experienced closed bands getting caught as well, but not as common as an open band.) So yes, some people do cut off leg bands in the name of safety.
All that aside, what are leg bands for?
Most are for identification with numbers and letters meaningful to only the breeder themselves, but at least, the breeder would be able to track hatch date and ancestry. Other times, leg bands are put on when going through immigration.
So consider that they are for identification. Ideally unique identification. If your bird gets lost, you might have a case for proof of ownership via that unique number (this is typically in the case of a closed leg band, which is only put on when they're babies). If you had a larger bird, an alternative to proof of ownership would be a microchip, which is a bit too big for a budgie to have. This ability to prove ownership is crucial if you have to bring the case to the police - sometimes people keep lost birds ("finders keepers" and all) and you might have found him but the finder might be unwillingly to return him to you. Without concrete proof of ownership, the police can do very little because it does turn into a "he said, she said" situation.
FYI, If you ever migrate out of the US, you would need either a closed leg band or microchip normally.
Now, going back to the proof of ownership, some people do keep lost birds and cut off leg rings to deny proof of ownership. Alternatively, kind people who want to return lost birds have no clue what to do with a unique identification number.
We have an ekkie, so I had him microchipped. He does have a leg ring (not a type found in the US, though), but it has my phone number on it. That way, if some kind soul finds him, should he be lost, they can just call me directly. Then I can prove ownership through the microchip.