The Freebmd records are very accurate and mostly complete for that era.
If you cannot find what you are after it generally means that the name is different (illegitimacy maybe) or the timing is wrong.
Remember that the age given in a census is x years and up to 364 days so the time of birth is likely to be 0ne year earlier than simple arithmetic would give you.
I tried for two of your family and found what seems to be them easily.
Births Q4 1871
Abram Silas Hardingstone 3b 35
Births Q1 1884
Abram Albert Northampton 3b 78