Right now it's more-or-less essential to take the zeroing perk as your active ability. It's so much more effective than any of the other options that it's negating what could be an interesting gameplay choice for players. Making the zeroing ability available by default and reworking the perk to be an advanced version of it could fix that.
The basic ability to choose a different zero (or to adjust elevation, which the perk more closely resembles) should be tied to the weapon/optic, not the player. All players should be able to do it on any weapons that have a real-world ability to adjust elevation (through flip sights, tangents sights, etc.) and on any weapon equipped with an optic that would be able to.
The current zeroing perk could be renamed to something like "Fine Tuning" and would give the player increments between the default ones, for a total of 5 instead of 3. For example, by default a scoped rifle may have options for 75, 150, and 300m; with Fine Tuning the player would be able to choose from 75, 100, 150, 200, and 300m. This would make it a valuable ability, but not so essential that every player would choose it. Some would be content with the default options and would use the opportunity to explore the other active abilities.
The idea of adjusting a weapon's zero could be preserved in the customization screen. There you could adjust the zeroing on weapons that can't be adjusted in the field, and for the ones that can you could adjust which option it starts on. I'm not sure whether this would work best as a standard ability or as one of the levels of the Fine Tuning perk.
On a related note, and even if you don't do any of the above, please make it so zeroing adjustments don't apply across all weapons. It's unintuitive and I've missed a fair few shots with a rifle because it slipped my mind that I'd previously adjusted a shotgun.