Does this math work? They are 6 over with bogeys and claw one back with a birdie but then have nearly 2 doubles.
Yes, it can work.
First and foremost, your handicap is not your average score. It's actually not even based on your score relative to par. It's based on your score relative to the course rating and slope. This converts your raw score into something called a differential.
But the first thing is that the handicap is not based on your
average differential. It's based on the average of only your best 8 differentials out of your last 20 rounds. So you really only play better than your handicap about 20% of the time, not 50%. The
average score for most golfers on 18 holes is usually about 3 strokes worse than your differential.
So
if the course rating were equal to par, a zero handicap would be expected to average about 3 over par. Which considering you've got 2.2 birdies (-2.2 strokes), 4.6 bogeys (+4.6 strokes), and 0.7 doubles (+1.4 strokes), according to Arccos data, a zero handicap averages 3.8 strokes over par.
Close enough for government work, given that I don't know the course ratings a "typical" Arccos zero handicap is playing from. For example one of our local courses, Tijeras Creek, from the back tees is par 72 and course rating of 73.7. So a zero handicap actually shooting even par would be a round that's 1.7 strokes better than their handicap, and ~4.7 strokes better than their average.