Speed-run of the theory, btw:
-Miles Naismith Vorkosigan has no notability in and of himself, he is only notable attached to his parents
-The only reason anyone would know about him existing is because of a birth announcement or other public notification issued by his parents, and thus he would be part of their bio
-Sometimes IRL these kind of birth announcements have middle names and a lot of info. Sometimes they don't. Miles, considering the circumstances of his birth, probably did not have one of these at the time he was born. It's unclear if he ever had one.
-Thus it's fully possible that no one on Barrayar who isn't aquainted with his family or knows his family well would have ever heard of him
-If they have heard of him, they know his name is Miles Vorkosigan, mother's maiden name Naismith
-Cordelia's father's name was Miles Naismith
-Miles Naismith's tragic death probably came with some publicity, including the fact that I assume it left no body
-Even if someone did know that Cordelia named her son Miles Naismith Vorkosigan, it's ridiculous to think that a mercenary admiral is a 17 year old nobody from a random planet
-This is before we get into language barrier issues, which the series mostly ignores. If we do take it seriously, I want to you to take a globe, pick a country at random, find the English wikipedia article on its current head of state, and then check the wikipedia article on its head of state in the official language(s) of that country. There will be differences. If there's a large wikipedian community that works in that language, there's going to be information only in that language that is not in English.
-Now let's assume that Miles's treason trial at age 17 made the news. This doesn't help the idea that this failson is a mercenary admiral, either.
-googling "Miles Naismith Vorkosigan" will get you, at best, "Cordelia Naismith married Aral Vorkosigan and they have a son named Miles"
-googling "Miles Naismith" will get you the admiral and old news articles about Cordelia's dad's death
-quick tell me Aral's middle name, tell me Cordelia's, tell me Gregor's, tell me Cordelia's mother's. Middle names are ignorable. Miles Naismith is Miles Naismith. Miles Vorkosigan is Miles Vorkosigan.
-but space google thinks that Admiral Miles Naismith, who is Betan, is Cordelia Naismith Vorksoigan's dad
-this is of course laughable, and it becomes a spacebook sign that you have to be careful when googling
-and then some folks are like "well, Admiral Naismith does seem to be doing good things for Barrayar some times" and then it's like "lol guess he's giving his daughter a birthday present" and no one actually believes it
-this is like how I met someone named James Bond once