Yes, indeed, Richard was determined to make sure the proper succession occurred. In fact, he was somewhat obsessed with royal bloodlines. It's why he got so fed up with his greedy commoner in-laws. The boys were declared illegitimate by Titulus Regius. This is a fascinating document all but destroyed by the Tudors until one was found in the Tower about 2 centuries after Richard's death. Until that time, no one knew the full story of Richard's ascension.
There already was a proper succession, which Richard was pledged on his life to defend. Not only did he not uphold his vow to defend it, he was the man who actively destroyed it, and for which he was JUSTLY DESTROYED. If anything he was killed too mercifully. He should have sat in the Tower for a few months and then been given the ax.
Titulus Regius was produced at Richard's behest; Parliament in those days did exactly what it was told. The production of the document long after Richard's death proves nothing whatsoever about Richard's claims. Among other things, it purports that Elizabeth Woodville and her mother used "witchcraft" to alienate the kings affections.
So, I can see why a Ricardian is impressed by it. But no one serious is.
As for "allowing" some of his relatives to live, my, that was generous, wasn't it? Especially considering that there is no doubt that he had the children he was sworn to protect murdered -- and since Edward was king, and Richard was not, that makes him a traitor in every sense of the word: to family, to country, and to oath.
And please don't start in with the usual Ricardian nonsense about there being "no proof" of Richard's regicide. In 1483 it was common knowledge they were dead; Mancini even recorded it contemporaneously in his diary. Richard himself responded to those rumors publicly. But what he did not do, because he could not do it -- was the one thing that would have set those rumors to rest for all time -- produce the children alive. That would have been impossible.
If Richard really believed his claims that the succession was illegitimate, why did he have them murdered? Bastards weren't ordinarily done to death. But of course, they weren't bastards, Richard wasn't a king, Richard's own marriage was of dubious validity, and he was a traitor, a usurper, and a murderer.