However, I have a birth certificate. Does that prove it is mine? I will argue with them.
You can argue all you like, but as soon as you obtain a birth certificate you are in fact effectively 'owned' by the state.
And any document issued by the state can be withdrawn/withheld/altered for whatever reason by the state.
(Remember the withdrawal of passports for not only those convicted of football hooliganism, but 'alleged offenders'? Or the withdrawal of driving licences from those alleged not to have paid CSA maintenance?)
That is because you are in fact what is known as a 'legal fiction'.
Look it up - it's most revealing!
Hence your 'legal status' as regards the right to be in the UK is not in your hands, even if you believe you can 'prove' who you are.
(And millions of EU citizens are about to find that out!)
The Windrush situation indeed shows that whether by incomptence or by design (given the warnings and info they had, the latter is the more compelling reason) it is an example of how the state can abuse your rights for whatever reason.
The last attempt at an ID card system was planned to facilitate the cross referencing of all government departments/public services so that at a push of a button your perfectly normal life can be disrupted or ruined.
And the best bit (as with most of government legislation) is that if they are at fault for a cock-up (or something more sinister) they can't be held to blame!
(Try looking up about those who had their motorcycle licence entitlement withdrawn by error on licence renewal, but had to re-take the test at their own expense)