Hi, your reply helps clarify the situation. Sounds like some other people are incorrectly identifying their ancestry, and they will not accept your evidence that they are mistaken. Arghhh. Frustrating for you - when you have done the legitimate genealogy, found the evidence, researched the connections.
Sounds as if they are misinterpreting and misusing the DNA findings. Lot of that going around.
The question isn't really about what you can or cannot prove re the babies. The question is really: Can these other people prove a given baby LIVED? There is no proof that it is the genes of one of "your" babies that has been passed down, and in fact you have evidence to the contrary. So aren't they the ones making unfounded assumptions, not you?
I've helped a lot of people research their family histories, and I am often astounded at how little grasp some show about how to interpret findings and evaluate evidence.
About this: You say "we family historians/genealogists DO have to make some assumptions, based on common sense and logic." Well, we form a working hypothesis based on common sense and logic. We investigate the hypothesis. We have to have a hypothesis, in order to find a research direction. I think that's what you do. We are each using a different term for the same process.
Have you ever read "Blink" by Malcolm Gladwell? He tells the story of the Smithsonian acquiring an artifact. An expert was led into the room to admire the acquisition, but when she saw it, she blurted out, "Oh, that's too bad!" Questioned as to why, she answered that she didn't know why she said that. She just had a quick gut sense that something was not right. After extensive investigation, of chemical make up or whatever, the artifact was found to be a fake.
Malcolm Gladwell calls this phenomenon "blink, or thinking without thinking." What seems like an instantaneous knee-jerk reaction is actually not lightning fast at all. "Blink" grows out of years and years of education and work and experience that give the expert a seemingly "intuitive" ability to perceive, which is in fact not intuitive but is instead the result of immersion. Gladwell estimates a person - no matter how bright - needs about 10,000 hours of immersion in the subject to acquire the necessary mastery.
I THINK you bring up all these babies who get baptized and then disappear because you sense a pattern of infant deaths that is hard for you to "sell" to others. You do have evidence! But you also seem to have an intuitive grasp of the pattern. It's tough for you without absolute PROOF these babies died. But again, I think it's the other folks who need proof that the baby lived. Just having the same name is not enough!
You aren't asking me or anyone else to evaluate the specific evidence; you wanted a general response. Without knowing all the details, I'm gonna roll the dice and gamble and put my money on you. You've done 60 years of genealogy in this surname, you did the homework, and my guess is that you evaluate and interpret the situation correctly.