That’s just not true. All engine oils need zinc. Zinc is part of the dry film lubrication that happens in all engines.
The rings need it. The guides need it. So it has really nothing to do with what lifter you use other than you can use LESS ZDDP with roller lifters and that’s only because the ZDDP is a consumable in the oil. If you start with less, you need to change the oil sooner with flat tappets than rollers because it gets used up faster.
So the amount of ZDDP (in PPM) is based on what lifter you use because it gets used up faster and not because a flat lifter requires more.