I'll assume this is worthy of a response. Suppose we look at family names rather than genes, assuming the name is passed down only by fathers. It is possible for a name to be bottlenecked at a single individual, even though there are many people alive and having children. It is possible, starting with a small population, for one name to become dominant. This can happen even if all the males of various names have children, since female children don't contribute their family names.
Is this likely? It depends on the size of the population and the kind of society. There are instances in recorded history where all young males in a population have been ordered killed. You may have read about some of these instances. People do strange things.
Simple version. There function "is a mother of" (or "is a father of" mutatis mutandis) is a contraction mapping. A mother may have more than one daughter but no daughter can have more than one mother (not even Heather.) Thus convolving (doing things more than once) makes the number of grandmothers smaller, etc. The end result is "1" for the "least recent common ancestor." (Look up "least recent common ancestor.")
Take all the women alive today (Please!--Henny Youngman) and consider their mothers, then consider their grandmothers, greatgrandmothers, etc. It doesn't take long to get back to only one person.
I don't see how your example addresses the facts of whether an individual or a group mutates.
Pitcairn island is an example of a few surnames becoming dominant and other surnames dying out.