Though death certificates are authoritative for the death, for birth, they typically rely on an informant. If the informant was the mother, she likely supplied correct information but if it was the housekeeper next door, maybe not so reliable. A birth certificate should normally trump a headstone while a death certificate might not.
In Ireland, as one example, there was a penalty if the birth was not reported within the required period and many parents would give a birth date later than the actual date but within the period that would avoid a fine so, for Ireland at least, I tend to favor the baptismal record over the civil record.
Keep in mind, even official records that are authoritative might be provably wrong. This is certainly the case for some birth certificates where DNA clearly shows that the reported father is not the actual biological father.
Long story short: a little detective work is needed but George is correct to say that marking a suggestion false is often the way to go.
