Yes it's always been that way in the UK. I think now the registrar observes the ceremony and then ensures the couple & witnesses sign the wedding registry.
Not any more, a local brother is appointed as the registrar and completes the register. There are certain administrative requirements such as the registration documents need to be kept in a safe, the "special ink" is used (£15 small bottle) but it has been that way in the UK for many years. I have been married over 50 years and it was that way even then.
George