My problem with this is that self ID basically already exists because organisations aren't asking for GRCs. My workplace celebrated this month's theme (LGBT history, IIRC) by reminding all staff that we can use facilities which match our gender identity.
I say, let them have GRCs for free, complete with a picture of a rainbow and space for some glitter. But change the law so 'woman' definitely just refers to 'adult human female' and, instead of 'allowing' single sex exemptions, list the spaces you'd expect should be that (toilets, changing rooms, hospital wards, prisons, support groups, etc). Mixed sex facilities can be given in addition rather than instead of.
While I don't have a problem with GRCs, it shouldn't be possible to falsify birth certificates.