Your father at age 6 was brought up by the OLIVER family - in Gray Creek Alberta.
He was raised by the family?
Does he know - why he was left with the Oliver family. Perhaps the mother was going to marry - or re marry ?
Note: The Depression - 1929 -> - may be the reason that the mother had to give up her son.
He was 6 years old - does he have any memories of his mother - perhaps a name - though children often are not aware of parents first names.
Was there an Adoption ?
Do you have the names of the Oliver family - the parents - perhaps their childen
Perhaps there was a family connection.
Your Father never had a Birth Certificate - ?
Have you checked with Govt of BC - Vital Statistics - in Victoria BC
The Mother or birth parents - may have registered the birth - which will have the mothers name / perhaps the fathers name - some info re the birth mother - eg - other children etc..
The Birth parents / or the Oliver Family - may not have obtained the Certificate of Birth - which is an extract of information taken from the Birth Registration.
though - without a surname - !!! might be difficult - don't know if they have the ability to check births for a particular location. -rather than name.
Was the birth at Gray Creek ? - If there was a hospital - There should be records of the birth - a Ledger- will hold information re patients etc - situations
If a home birth - there may be records -from a midwife - or doctor.