That's not how it works, the chances of two people matching as identical (on the SNP tested by 23andme) by random is astronomically low - orders of magnitude higher than the total number of humans who've ever lived.

Generally at a 7cM overlap you have 50:50 chance whether you're related to that person or not. Anything over say 150cM you're virtually certain to be related (ignoring endogamous populations). At 3,400cM (full-match) it's not even worth considering randomness.

You’re assuming a lot with those statistics.

Paternity tests involve random ~50% of your dna. Sperm samples are generally contaminated and thus less clear cut. But, people’s relatives have been used in the past for these tests dropping the bar even further.

However, being related genetically is not enough. Some people have twin siblings they don’t know about pointing out being related is not nessisarily mean you know anything about the other person allowing for false positives even at 3,400cM.

Unlikely sure, but harm comes in many forms. Saying I can’t think of anything is a long way from saying it’s safe.

Those aren't false positives in that they represent real genetic relationships (i.e they're identical by descent), as the database size increases you don't increase the false positive rate in any meaningful sense at higher match levels.

If you've got a identical twin then you're right that their child will be genetically indistinguishable from your child. But that's down to the nature of genetics rather than anything to do with the reliability of testing or database size.

False positives don't necessarily mean the test was executed incorrectly. It often means you are testing for something closely related to the information you want.

If a test failed because _ then it failed independent of why it failed. If your long lost twin causes you to be involved with a paternity suit or homicide investigation then that's both a false positive and a significant problem.

Why couldn't you run those tests several times?

These are not false positives, because you ran the test wrong.

Suppose a male with red hair, green eyes, and AB+ blood actually has those traits they are just not enough to unlikely identify someone. Adding more genetic traits on it's own is not enough for example you have identical twin or even triplet separated at birth. The core issue is DNA marks are not independent though people often assume they are.

Society will have to adapt to a better understanding that this is true. Keeping one’s DNA secret is not going to be a useful defense against ignorance of how DNA works.

Worked for the last 10,000+ years and still works today.

Sure, at some point in the future it might get fixed or always fail, but until then you need to deal with today not some mythical perfect future.

The last 10,000+ years has not been static from my perspective. Blood testing came along. Limited DNA testing came along after that. Perfect DNA testing will be here soon. The technology won't care how you feel about it being used. I'm not imagining a mythical perfect future. I'm imagining a very real future that we can't avoid and need to deal with more pragmatically.