But you don't have to broadcast your tail number to prevent a collision. It makes tracking by ATC easier since the tail number can be traced back to a flight plan, but if the sole reason is to prevent in-air collision, a radio beacon without tail number could achieve that.
A tail number also makes it easier for ATC to know who to give runway landing clearance to, doesn't it? They can't just say "you airplane over there, yes you, you're clear to land".
Whats broadcast is not the tail number but a ICAO ID assigned to the aircraft, that database is public. The FAA can assign anonymous IDs to aircraft that need privacy for some reason and still be able to identify which aircraft it is in their systems.