Actually it seems like I was indeed in the grip of a myth. Sorry all.
I wouldn’t recommend studying for a PhD in pure math. You will get some sort of decent job, but there isn’t a lot of opportunity, like in some other academic fields. Plus the competition is tough, because people studying math are all smart. A lot of it is like how many angels can dance on a head of a pin. IMO, math applied to science and engineering, numerical analysis, statistics, and operations research are more interesting, as well as having better career prospects.
I have two math major kids. One is now in the second year of his Phd (pure math). The other is in his third year undergraduate (probably CS double major, he’s really done with his math degree and core requirements already and could just graduate if he chose). Both will find their way. The older one knew he’d be a math major when he was quite young. The younger is pretty similar in interests. Math makes them happy.