This is not a problem, this is outlook working as designed. When you use anything except your default main calendar, it is not going to update the default calendar. Of course it updates the calendars of the invitees, the invitation goes to their default calendars. If the organizer wants their own default calendar to also be updated then they have to set up the invitation in their default calendar. That’s just the way outlook was designed.
I would suggest rather than sharing a second calendar in this person’s outlook that perhaps they have a resource or room mailbox created instead and invite the resource with each meeting request. Then everyone can check this shared resource calendar and all will be updated correctly on their own calendars.