Why doesn’t the iPhone Fitness app track activity while golfing?

The motion of a golf bag is very different from that of walking.

The iPhone expects to be in a pocket during a walk, as such, it can measure the swing of your hips and calculate your step cadence/pace to infer that you're walking, and log distance/steps accordingly.

In a golf bag there is no swing (no pun intended) or pace... just linear movement, so the phone can't determine that you're walking alongside the push cart and log a workout.

You might have more luck if you push the cart in spurts, just like your arm would swing normally during walking, vs. pushing the cart at a steady pace, but that's all I can think of off-hand.

It’s tracking what it’s attached to. Not carrying the iPhone and not wearing the Watch won’t particularly help with tracking your own physical activity, and probably also isn’t accurate for tracking the physical activity of the pushcart itself, either.