You seem to be answering your own question. Genealogists cannot establish paternity. Geneticists have in the right circumstances established non-paternity and likely paternity.
Historic genealogies are largely based on faith - we normally assume both husband and wife were the parents of a child unless paternity is publicly contested by the father. We also normally assume that acknowledged natural children were likely fathered by the man who officially acknowledged them. It is a matter of faith because in most cases certitude based on hard evidence remains an elusive prospect.
Finally, yes, it is very likely we would have never heard of Charlotte and her illustrious paternity had the Prince also fathered a child born in wedlock.