Those who were not heiresses
My paternal great-grandmother, Sarah Ann Bartholomew, was a genetic heiress. I suspect she was the carrier of the red hair that my grandfather, my father, I and and my eldest son have inherited and she appears, from her recently discovered photographs, to have passed on her good looks to some of the females of my and the next generation.
But Sarah had nothing more. Her father when she was born was the ostler at the Star Inn, Newmarket though later in life he had his own Inn in the village of Ashley a few miles away. Her father left a few hundred pounds but I don't think any of that went to Sarah, much as her husband was the executor for his father-in-law. And certainly her father had no coat of arms.
Sarah had at least two brothers but, save one, they disappeared early, almost certainly youthful deaths. Her brother William survived Sarah by over twenty years but he had no children and left his assets to his sister Eliza.
So Sarah's brother died without children, she had a sister, they were co-heirs to this eldest brother William, though Sarah was but an heir in her issue, she was not an heir while she was alive. Sarah then was a pure genetic heiress, carrying no property or arms to her offspring. She was not an heiress at all in the modern sense of the word.
There are many others like Sarah, genetic heiresses but who passed only their genes to their offspring, no wealth, no arms and certainly no title.