Your family may not have been here before 1700.
Birmingham was a very small place in 1700, before they built the canals.
In fact Coleshill, a village to the East of Birmingham, was BIGGER than Birmingham back then (hence the large spire in Coleshill church)
The building of the canal system, that centred around the Birmingham area, brought thousands of people from the surrounding areas and Birmingham grew very quickly as a city.
Your family may have been "country folk" living well outside the Birmingham area in 1700.