What I gathered was that he was far too intelligent a man to worry about anyone threatening his position as King of England. I would say that the state of the country when he inherited was dire, and required an extraordinary person to deal with it - certainly an achievement overlooked by most historians who have concentrated on monarchs who have either a bloody outlook or achieved high esteem on the battlefield.
Henry did neither of these for he didn't have to - using others to do the work for him. In essence and given his tentative situation he 'should' have turned out a leader with a great deal of blood on his hands.
He was ruthless in his own particular way unlike Bloody Mary: foreign policy, national policy, law, control, army (which he cleverly 'rented' from nobility in order to consolidate his weak financial state), set precendents throughout Europe and so forth.
I do believe he would have overcome the challenge had those two princes survived. It just so happened that the princes had disappeared, though it wouldn't have been a concern to him had they lived as he was in a league of his own at the time. Just as he had dealt with the problem of letting both Simnel and the Earl of Lincoln live, he couldn't be touched.
I sound like a fanboi but he really has been underestimated as a monarch - forget Henry V at the Battle of Agincourt or Richard the Lionheart for they were only one hit wonders

I should also add that this is the time that is credited as the death of chivalry (though in my view the death of chivalry occured the moment those cannon were fired at the Battle of Agincourt/Crécy). Henry Tudor was an honourable man and the Simnel/Earl of Lincoln fiasco only serves to reinforce this notion. He wouldn't have stooped so low as to have those princes eliminated. That's my view anyway.