In your context, I wouldn't understand that it means he has more of an advantage than someone else. Rather, that he's the one in charge: 'He heads up the family business'

It also implies 'heads up', which is stronger -- see the explanation below:

What Does Head Up Something Mean? - Writing Explained
https://writingexplained.org › idiom-dictionary › head-...
That said, to head up something can be conjugated as heads up, adding to the confusion between the phrases. *He heads up the task force on foreign relations.

The saying 'Heads up' - meaning and origin. - The Phrase Finder
https://www.phrases.org.uk › meanings › heads-up
... Another meaning, which has no connection with one's own head is when heads up is used as a verbal phrase; for example, "We have a new boss. He heads up the whole organization."