I've never understood it, either, particularly since the Mayflower wasn't even close to the first group of European or English settlers in what later became the USA and half those who landed died the first year (further reducing the pool of available ancestors). It's kind of like all the people I meet in Texas who claim to have had ancestors at the Alamo...I don't challenge them, and it may well be true, but given how that battle ended I do point out that it's good that their ancestors were conceived beforehand and kept offsite. I did meet somebody who claimed to have an ancestor at the Alamo, but on the--ahem, winning side--and that always gave me a chuckle. No idea if it was true.
Fpr the record, my Great-six-times-removed grandparents were in the Republic of Texas by 1835 but were nowhere near the Alamo or any other significant historical event. They were farming and apparently trying to keep their heads down.