I am blown away that this seemingly simple functionality is missing - There is no simple way in setup to send an email to a member after you add them to a case team. I was resigned to doing a more complicated route (a flow or a trigger that fires off an email on create of a CaseTeamMember record) but that object isnt even an option to select in either a flow or trigger! It looks like you can create flows/triggers on campaign member and opportunity team member so I'm not sure what the reason is here.
I know I can query the CaseTeamMembers table in Apex but I have no way of automating this to send the email on create... does anyone have any suggestions?