Often?
William I didn't kill Edgar the Atheling.
Henry II didn't kill Stephen's son.
Edward IV didn't kill Henry VI until pretty late.
Trying to think of other examples for England post-Conquest (which I know the most about).
It would make far more sense in a medieval context to have Richard and his ancestors as loyal vassals than to go around killing them all.
I can't speak for other cultures like say Japan, but medieval Europe seems to have been fairly reluctant to go around killing people just because they might be pretenders as opposed to imprisoning them.
Byzantium used blinding, but that's more Byzantine preference for blinding as a means of getting rid of someone.
I'm not sure those examples are very good ones to use, Elfwine.
In the case of Edgar the Ætheling, William kept him in his household as a hostage for a year or so, until he got himself involved in a rebellion against William. When William found out, he fled to Scotland, where he was involved on numerous occasions with sponsoring more rebellions. When William attacked Scotland, Edgar had to flee to Flanders. Some eight years after Hastings the King of Scotland persuaded him to make peace with William, but William's poor treatment induced him to go to Italy to make a new life for himself there. When William died, he joined William's son Robert and one of his principal advisors in helping him to attack and dethrone William II. This was hardly a harmonious relationship between deposer and deposee.
In the case of Henry and Stephen's son, neither side fully won The Anarchy, so the adoption of Henry over Stephen's children was a legal compromise. There was no deposition - Stephen was allowed to keep his throne - so there was technically no reason to do away with rivals, not least because doing so would probably be seen as an invalidation of the treaty and would lead to a resumption of The Anarchy. On top of this, Stephen's line was weak anyway - his eldest son had just died, leaving no issue, and his only remaining (legitimate) children were his daughter Marie, who was currently a nun and therefore hardly a strong choice, and Stephen's youngest son William who was barely 16 or 17 and who promptly started scheming to have Henry II murdered and had to flee to Normandy anyway. Given that his own father had turned his back on William and he had never been crowned - not even crowned as heir, as Kings of England liked doing to their eldest children in this era - it's likely that he wouldn't have been believed to have had much support and therefore would've been treated as insignificant.
In the case of Edward, it took him five years after being crowned the first time even to get possession of Henry, who was in hiding in Scotland, and then he promptly kept him locked up in the Tower of London. After Henry's restoration and then re-deposition, Edward did in fact have him killed.
So to summarise: in two cases, the deposed King wasn't kept as a loyal vassal but was in fact kept under close observation under lock and key. In the other case, the "dethroned" man was allowed his own land but lost it in months after plotting his own overthrow of the new King, and ended up fleeing England. In all three cases, the "deposed" were actively involved (maybe not intentionally in Henry's case, but that was a symptom of his mental condition) in attempted coup d'etats. In short, keeping a deposed man alive really was an inviable state - if you tried it, you knew that you were going to be backstabbed. After a while, by the 14th century, they just stopped trying to keep them alive.