"The further back you go, the less likely a person has descendents living today."

What is important is the first few generations and also the diversity in those generations [your children and their children] if someone 100,000 or 10,000 years ago, had 4 children and these children had 4 children, then their would be a near certainty of survivial to the present. If instead you had an average of 2 children each [slightly less than replacement- if it was average rate it would cause declining total populations] you would still have a high chance. If someone has a known surviving descendent after 1000 years then it's very likely that you not only will have descendents surviving 10,000 years after this but most of people living at that time will be your descendents. It's simple math: e.g. 2 to the 100th power.

"Everyone living on Earth in 10,000 years will be the decendents of only a handful of people alive today."

Nonsense. If you are a grandfather with 2 children and 4 grandchildren who are having children or have children, then you will have descendents 10,000 years from now: 2 to power of 500 [assuming nice round an average generational time of 20 years]. If on other hand you are great grandfather with 1 child and 1 grandchild, and 1 great grandchild, then your chances are lower, but if any of your later descendents has 2 or more children and thereafter it becomes the average then in 10,000 years almost everyone living will be your descendents [if this is typical, then total human population would crash, btw].

If you have 1000 couples each only having 2 children for say 1000 years, and you had 1000 couples each having average of 12 children for 1000 years. In 10,000 years both the 2 children group and 12 children group will be the descendents of almost all people on earth, but far more of the DNA of the 12 children group will be part of that population [assuming 12 children group don't exclusively marry siblings or cousins- diversity *could be* more significant then sheer numbers of children from a particular group]