Hey Henning, Very intereresting read. I knew about Yoshitaka (Gigo) I think we all do because of his involvement in Karate. However I only knew a bit about Giei. the information I read was that Giei did teach Karate with the Shotokai however. I have read this at several online sources yesterday, one of them being an official Shotokai site.
I would be very interested in seeing the sources for your information about his other sons. From what I read here it sort of states that the other sons basically did not learn Karate, one because he had passed away young. Yet Funakoshi himself says that his "Sons" learned Karate from the masters and they were spoiled with candy and sweets.
I guess I am just having issues balancing the common thoughts on them with the information you present. Can you provide sources (not formally just a idea of where you found this out from).
Thank you in advance!