On page, https://www.zabbix.com/documentation/4.0/manual/api/reference/host/object, inventory_mode is shown as part of the host object, when a host.get doesn’t return it as a base part of the object.

Under the host inventory table on the same page, it isn’t listed as part of the inventory object at all.

I haven’t found anywhere in the manual where it clearly states that to retrieve the value, you look in the inventory object, but to change it you use host.update on the base object.