Basically the title. For my mob class, I have the following 2 methods overridden:

   @Override
    protected void populateDefaultEquipmentSlots(DifficultyInstance difficultyInstance) {
        this.setItemSlot(EquipmentSlot.MAINHAND, new ItemStack(ModItems.COMICALLY_LARGE_SPOON.get()));
    }

    @Nullable
    @Override
    public SpawnGroupData finalizeSpawn(ServerLevelAccessor serverLevelAccessor, DifficultyInstance difficultyInstance, MobSpawnType mobSpawnType, @Nullable SpawnGroupData spawnGroupData, @Nullable CompoundTag compoundTag) {
        SpawnGroupData spawnGroupDataToReturn = super.finalizeSpawn(serverLevelAccessor, difficultyInstance, mobSpawnType, spawnGroupData, compoundTag);
        this.populateDefaultEquipmentSlots(difficultyInstance);
        return spawnGroupDataToReturn;
    }

I've already tested the item separately, so I know it works fine. However, I still can't get my  mob to spawn with the item equipped. Is there anything else I have to do for this?

The method is called. I debugged the code when I used a spawn egg for my mob and stepped through the code, and it did call setItemSlot. I would hit the breakpoint in the debugger both using the spawn egg and using SpawnPlacements.register.

I'm not seeing the item in the mob's hand when it spawns.

That might be it. My Renderer currently extends MobRenderer. It looks like the following:

public class MyMobRenderer<Type extends MyMob> extends MobRenderer<Type, MyMobModel<Type>> {
    private static final ResourceLocation TEXTURE =
            new ResourceLocation(Main.MOD_ID, "textures/entities/my_mob.png");

    public MyMobRenderer(EntityRendererProvider.Context context) {
        super(context, new MyMobModel<>(context.bakeLayer(MyMobModel.LAYER_LOCATION)), 0.5f);
    }

    @Override
    public ResourceLocation getTextureLocation(MyMob entity) {
        return TEXTURE;
    }
}

I'll try extending HumanoidMobRenderer to see if that works.

Thanks for mentioning the renderer. Extending HumanoidMobRenderer worked since it has logic for rendering items in hands.