To the seccond idea: I would not inherit the avator of the organization 1:1, but perhabs it would be cool if the logo would be in the background of the initials. Otherwise, if every user of a organization has the same avatar it could be a little bit confusing.
This would be even more awesome, now that the organisation icon is displayed at the top of a ticket.
In addition, it can be automaticly filled by the Clearbit-integration (if they provide logos, I donβt know).
And if domain-based assignment is used our Image Service can be used as well. The one that tries to find a login logo based on the e-mail-domain of the administrator in the setup process.