5% of males within the old Mongol Empire have been shown to have nearly identical Y-chromosomes – it’s only a (very good) assumption that this is the chromosome of Temudzin “Greatest of Rulers”. With this huge agnatic descent, Temudzin must be ancestral in some way to 99% of Central Asians, any exceptions being recent immigrants.
Edward III is not a similar story. While simple math might suggest he has many millions of living descendants, class barriers may make the actual number much smaller. Certainly most American genealogists are unable to find their own proven line of descent from Edward III.
Reasons for the greater procreative success of the Mongol Khans are well known:
Five hundred wives and concubines composed the harem of Genghis
…
Kublai Khan’s underling, the celebrated Italian traveler Marco Polo, wrote that each year the emperor took 30 additional virgins to be his concubines from a province renowned for the beauty of its women.
This may be applicable to Europe’s “Dark Ages” but isn’t universal. Many Chinese allege “proven” descent from Confucius, who lived 2500 years ago. Aristocrats of ancient Persia, Greece and Rome kept records; the problem is extending those genealogies to the present. It might be doable if a single blood-line prevailed as rulers throughout such a long period; Ireland’s O’Neil family may come close but I don’t think historians extend it much before Niall Noigiallach (born 4th century).
The Saud claim is, AFAIK, just fiction. The King of Jordan, on the other hand, shows on his own website a detailed agnatic lineage back to Ali ibn Abu Talib (who married the daughter of the Prophet). I don’t think that line has been de-bunked (but doubt it’s been bunked either).
For long lines, a question that arises is how thoroughly must each link be known to be valid? If some ancient’s mother was “almost certainly” a Frankish Princess can that be counted as a link, despite that details are unknown?
For example, Alfred the Great’s grandmother Redburga was described contemporaneously as “regis Francorum sororia”, but no details are confirmed. Yet all reconstructions make her a g-g granddaugher of Charles Martel. Would you accept that all descendants of Alfred are therefore proven descendants of Charles Martel?