Cherokee people are very happy to share their genealogy. What we don't like are people who claim Cherokee ancestry without documentation. There are Cherokee citizens, members of one of the three Cherokee tribes, and there are Cherokee descendants, ineligible for tribal membership, but with ancestors who were Cherokee. Whenever we meet a new Cherokee person almost the first question is "Who are your family?" because we are so interrelated. Cherokee people never lived in a vacuum. Even if some individuals left the tribe or refused to sign up on the Dawes, other members of their family did enroll so we know who they are. The Dawes Commission attempted to track down every single Cherokee person on the 1896 Cherokee census for the Dawes. Those who hadn't enrolled were often signed up by other family members. The few who didn't enroll can still be found on that 1896 census and other records, so again, we know who they are. The "Too Lates", Cherokee who moved to Indian Territory too late to be included on the Dawes still filled out applications and are listed on the 1900 U.S. census in Indian Territory, so we know who they are. There are similar cases with the Eastern Band; there were Cherokee descendants who weren't eligible for citizenship, but we know who they are from their applications and the applications of family members who were enrolled.
Most of the 19th century rolls that included Cherokee east of the Mississippi involved money so instead of hiding, people were eager to sign up. Yes, the Cherokee outside of North Carolina who chose to give up their Cherokee citizenship and remain in the East usually appear on the U.S. Census as white, but they still can be found on the Siler/Chapman and Hester Rolls. The Federal government (and people's neighbors) didn't forget who was Indian and they continued to push them to move west. The Feds continued to pay transportation and subsistence for families who moved through the 1880's. The South was a pretty racist place, and after the Civil War many Cherokee who had initially stayed behind went west and regained their citizenship. There are thousands of pages of Cherokee records of those citizenship cases; again, we have family trees, lists of parents, siblings, children, and other relatives whether the person was approved or not.
The Eastern Cherokee (Guion Miller) roll is the best source for information on unaffiliated Cherokee descendants. Tens of thousands of people who claimed a Cherokee ancestor applied and Miller followed every claim. If he couldn't find a Cherokee ancestor, then the odds are about a million to one there isn't one.
Lots of people whose ancestors never lived in the Cherokee Nation claim to be Cherokee. The Cherokee claimed a large territory, but their homes were all in a relatively small area on the southwest side of the Great Smokies along the Tennessee Rivers, and in northwest Georgia, never in Virginia/West Virginia, Kentucky, or east of the mountains in North Carolina. The Internet trees claiming Cherokee ancestry for Phoebe Crews are made up. She and her family lived in Person County, North Carolina, never in the Cherokee Nation. She may have had a Native ancestor, but there is no documentation which even suggests that is correct, and if she did, it was from a tribe in the Person County area, not Cherokee.
For a time, the state of North Carolina claimed that the only remaining Indians in that state were Cherokee and they listed the Lumbee as Cherokee. That does cause confusion for folks tracking their North Carolina Native ancestors, but the Lumbee also have good records going back before the Civil War.
Are there people whose ancestors married a Native person back in the 16- or 1700's whose identity has been lost? I'm sure there are, but there is no way to know who that ancestor was or to claim them as a member of any existing tribe. When it comes to being Native American, it's not who you claim, it's who claims you.