We know that’s how it works, they don’t.
It’s fairly easy to make that connection if you know the man’s connected to the Heterodynes, which they don’t.which blows his cover even if they don't make the fairly easy connection that he is a Jaeger.
Everyone knows what Jägers are like and Higgs is about as far removed from their image as you can get.
Hell, even if they figured out he was impossibly old, the obvious culprit for that would be Albia, not the Heterodynes.