Zddp is a high pressure lubricant. Stiff valve springs and super/turbocharging are where a lack of it will cause issues.
After a modest engine is broken in, it probably would live a plenty long life at the standard 800ppm most modern oils have. A super high lift solid cam engine would probably need more. A circle track engine could probably use more just due to the longer times spent higher in the rev range. Necessary? Maybe not..
The thing is, if you change the oil often enough, even pretty radical engines would probably live with low zinc. It gets "used up" during the high pressure cycles, and is slowly depleting. If you change oil each weekend, you're unlikely to run out.
But put in too much zddp, run it with a shitty hydraulic cam with puny springs, rev the engine in your driveway twice a year and put it away at 95F coolant temps, you're likely to have more issues than the guy running whatever was on sale at Walmart. Then again, if changed super frequently, one probably wouldn't have many issues other than smooth bores and higher oil consumption.
Everything is relative and application dependant.