King's Landing? Do you NOT remember what happened to Ned
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King's Landing is a HORRIBLE place for Jon to go. HORRIBLE. Jon needed to be out of sight, out of mind, far away from people like Varys, Tyrion, Tywin and Littlefinger who would be asking all sorts of uncomfortable questions.
Ned let him join the Night's Watch because it was the best place for Jon to go. When you join the Night's Watch, you forsake all claims, all inheritances. Jon would never be safe doing anything else. If he was a Knight, a Master of Arms, a Smith, a Maester, he wouldn't have been safe. If people found out about his heritage, somebody would want to use him to stage a rebellion and/or the Baratheons/Lannisters would want him dead. The only safe choice for him was the Night's Watch or to go far across the world like Dany and Viserys.
Remember Aemon Targaryen, the old guy that Jon took care of at the Wall, that's exactly the reason he joined the Night's Watch. So his brother could become King. If he did anything else, chose any other option, people would keep using him as a tool against his brother. By joining the Night's Watch, he gave up all inheritances, all claims, all rights.