After Worf chose to undergo the surgery to replace his spinal cord, he asked Counsellor Troi to look after Alexander in case he didn't survive. The fact that his human foster parents were getting up in age is understandable. However why didn't he entrust his brother, Kurn to look after his son? Kurn would definitely have made sure he had a Klingon upbringing.
I think I agree with you that Kurn would have been a good choice and one that Worf would have probably leaned toward, but just from a logistics "it's only a TV show" asking Troi was more dramatic. Bringing in Tony Todd for one scene wouldn't have made much sense.
It was eluded to that Kurn had children so having Alexander go live with them would have been logical. In "Firstborn" K'mtar tells Alexander that his uncle (assuming he means Kurn since he mentioned his name severl times in that episode) has a big house next to a lake and that Alexander had cousins there. He later says that Kurn has no male heir, implying the cousins must be girls. Which makes your point, Timo
, what the heck happened to his family after Worf screwed him over?