Norman Osborn's purpose, as you can read in the classic stories and later years (even a reason of contempt between Lee and Ditko), was not to be just anybody, but to be a powerful rich man that craves power and belittles everyone else unworthy in his eyes.
The Family angle is more recent, but not unwelcome. Norman sees himself in Peter, and not much in Harry.
Harry Osborn was not strong because he folded under his father's abuse, and Norman thinks himself stronger despite the abuse he himself had suffered.
Father of the Year, right there.
In the Ultimate Comics, Norman even associates Peter's worth to be because of Norman himself.
Peter acquired powers because of a genetically modified spider from Oscorp, and Oscorp is Norman's company, therefore Peter's greatness is because of Norman, and so Peter himself is Osborn's.
Peter is his 'heir' because, to Norman's eskewed mind, Peter is his creation, his property, one that should obey him or be punished in return.
The big twist in the (badly written and badly drawn) Ultimate Sinister Six Comics is Osborn's rampage over NY to complete his team with the help of his son, and the reveal that he meant it to be Spider-Man, not Harry.
(I assure you, despite what you may think, that's supposed to be Peter Parker.)
Mainstream Peter was always someone Norman felt worthy of power, but hates because does not use this power to improve himself, but the world.
And the current problem is that people like Wells and many others always try to see Peter Parker/Spider-Man in a very focused manner:
Peter is powerful and smart, so he should use these qualities to better himself. His great powers should serve himself.
But Peter is not an industrialist.
Peter is definitely not Norman Osborn.
And Peter sure as heck is not the bad guy.
Unless you are Zeb Wells and modern Spider-Man writers.
Then Peter Parker can be all those things, and less.
