"it is still possible to obtain a Birth Certificate for anyone with minimal effort"
Freely available to anyone who asks and pays the fee.
"and then use this to request a UK passport "
The method used should have been sealed in the 1960s. After all it's the registrar of Births _DEATHS_ and marriages, so it's not as if the relevant disqualification document is filed in another government department.
(FWIW, many countries _do_ tag records with a death date specifically to ensure that ID documents in the name of dead children can't be obtained. The UK seems to think this is too hard despite it being a known vulnerablity long before Frederick Forsythe wrote about it.)