That's what some lobby for. Some want self ID with no medical checks. This view has a point - now that being transgender is no longer considered a mental illness there's not really a reason to require doctors to sign off on a GRC. (Nor for any form of NHS treatment, if it's not an illness, but that's another argument).
I think it should not be possible to alter a birth certificate.
Beyond that, if we were very clear on sex and the difference between sex and gender then they could hand out grcs like sweeties and it really wouldn't matter. The problems are birth certs being changed and the utterly ludicrous elision of gender into sex.