Landlords can accept a birth certificate. The are legally obliged to check paperwork: https://www.gov.uk/check-tenant-right-to-rent-documents and not being immigration officials they are often confused about what they can accept and what they should check.
Employers can accept a birth certificate plus some other documents.
It is possible to open a bank account without ID, but its not standard and will be more difficult: https://www.fca.org.uk/consumers/opening-bank-account
Lots of other things come up. You need photo ID to sit GCSEs as an external candidate, for example.