Actually, the rumors (and there were really very few) simply suggested that young Edward may have died - not that he was murdered. There were some notions floating around that he was not in good health. Just as a point of fact, Richard was no where near the Tower during the time it is said that they disappeared. He was on a progression throughout England.
Happily, several pretenders showed up claiming to be the two brothers, making stingy, nasty, cowardly Henry VII’s life a misery!
If Edward died, then his brother Richard would have inherited his claims to be the “legitimate” Yorkist King.
I think the history is very clear that RIII’s main problem was his own supporters deserting him or switching sides, some of them on the field of battle itself. And a prime reason for that was the rumors that he had killed the Princes.
On a practical note, none of these kings of the period had any greater or lesser “right” to the throne than any of the others. It was a clearcut power struggle, that was all.
Of course, a king who came to power by these means soon found that his throne was insecure and that he could be overthrown by exactly the same methods. A good example of poetic justice.
Henry VII was actually a reasonably good king, as compared with other kings, admittedly not a particularly high standard to beat. He mostly kept England out of foreign wars. He was “oppressive” to the high nobility that threatened his throne, though to be fair the high nobility needed to be taken down several pegs. They had, after all, put England through several decades of horrible civil war pursuing their own interests.