Hunt: Showdown does it best, at the equipment screen you should be able to modify the ammo you want for your weapon to use (and pay extra for it if they want to go that route)
Ammo options could include:
Dumdum/HollowPoints that cause bleed/add bleed stacks on enemies - but weaker against armor
Incendiary ammo self explanatory - Also weaker against armor
Full metal jacket - Armor piercing ammo
etc
They would be more likely to add new weapons / weapon variants that can load special ammo (such as the shotgun) instead of introducing an entirely new mechanic.
The shotgun’s implementation of specialized ammo is really lackluster, they could have done something way more interesting than just horizontal spread. I would’ve preferred the baseline shells to have more spread, and have the option to load solid slugs to deal with tougher or longer ranged enemies.