You seem somewhat surprised that trans people can change everything but their BC so easily.
BC changes also used to be self ID before April Ashley’s husband got a divorce on the grounds April was born a man.
Anyone could simply go to their registry office where their birth was registered and ask for it to be amended.
The government was taken to to European Courts in the 90s because we were unable to make these changes, where they unsuccessfully argued that as all common forms of ID can be easily changed, the BC did not need to. This led to the GRA in 2004.
The trans community, and indeed most politicians have been telling you for the past two years the UK operates on a self ID basis and has done so pretty much forever.