"SaddleItem" is un-modifiable using "/data" unless for adding a single saddle on horse, donkey, mule, skeleton horse, zombie horse and camel. This is different from "body_armor_item" where you can literally put whatever item you want as long as it's an item that's exist and in the stack range.

Step to reproduce:

Step 1: Summon a horse, donkey, mule, skeleton horse, zombie horse and/or a camel, then tame them, then you could add saddle or not.
Step 2: Type: "/execute as @e[type=<horse/donkey/mule/skeleton_horse/zombie_horse/camel>,distance=..10] run data merge entity @s {SaddleItem:{id:"minecraft:brain_coral_block"}}" (Brain coral block is just an example item, it could be whatever you want) -> The horse/donkey/mule/camel still don't have brain coral block in their saddle.
Step 3: Now you have to add saddle, then type "/execute as @e[type=<horse/donkey/mule/skeleton_horse/zombie_horse/camel>,distance=..10] run data merge entity @s {SaddleItem:{id:"minecraft:air"}}" -> The horse/donkey/mule/camel still have saddle even though it's a legal item in the saddle slot.

Note that the "/item" command can turn saddle into air on the saddle slot, but it can't turn saddle into anything else.