My "gut feeling" is that eliding certain words from stock phrases is more common than one might think. Without having researched this gut feeling, I can at least give some examples of how stock phrases can "shrink." Take two wrestlers talking to one another:
"You have a twenty-pound advantage on me" versus "You have twenty pounds on me."
Here, advantage is elided. Or to stay with your "head" example, two weight-loss dieters who started losing weight at different times are talking, and one says to the other:
"You have a 20-pound head start on me" versus "You have a 20-pound start on me."
Or how about two sophomores who decide to see who can graduate with the higher grade point average (with 4.0 being the highest possible GPA):
"But right now your GPA is 3.85 and mine is only 3.80. You have a [point zero five] head start on me" versus "You have a [point zero five] start on me."
To switch things up a bit, it's not unusual in American English to elide words from other stock phrases, perhaps not in the same way as in your example, of course, but as a sort of shorthand:
"You know what they say: 'Hindsight is 20/20'" versus "You know what they say about hindsight."
Here, the speaker knows the listener will supply the missing "is 20/20."
Or,
"You know what they say about the early bird" versus "You know what they say: 'The early bird catches the worm.'"
Or,
"On Tuesday, Fred did an armchair-quarterback analysis of Monday's football game" versus "On Tuesday, Fred did an armchair analysis of Monday's football game."
And so it goes. Which reminds me . . ..
Person A: "How's it going?" Person B: "It goes" versus "It's going well, thank you."
And so it goes.