I'm not convinced Henry did either, though I don't think it's as unlikely as it seems... I'm not convinced the princes vanished when we think they did. We know they went into the tower in June and the last record of them is in August when someone writes that they were seen on the battlements practising their archery.
But just because that is the last record of them doesn't actually make it the last time they were seen. It just means that it was the last time somebody wrote down that they saw them - and not even necessarily that - it's actually the last time someone wrote down that they saw them and that document survived. Lots of letters and diaries etc could have held similar information and simply been lost to posterity as most letters and diaries are.
Though there is no reason in a semi-literate society that there would be masses of written evidence for what two disinherited boys being kept quietly in the tower are up to. One of the problems with the written record is that it doesn't tend to record the mundane, obvious and everyday. And things that are a record of that (laundry lists etc) don't survive because who on earth thinks to keep their laundry list for the historical record?
I remember one of my lecturers at uni pointing out that her diary was full of important meetings she had to attend... but there was never anything written in for half 3 on a weekday - the time she went to pick up her children from school. She did it every day so she didn't have to write it down to remember it. But as far as the historical record is concerned there is just a blank nothingness everyday for her at 3:30 when she could be doing something important but wasn't.
The people in charge of caring for those boys weren't going to write down that they had seen them because... well, obviously they had!
They don't actually vanish from existence. They vanish from the historical record. But as their importance has been removed - well, why wouldn't they?
I've said before - there was no rumour the boys were missing at the time, so nobody was particularly looking for them or taking note of them. After Richard is crowned, Elizabeth Wodeville comes out of sanctuary, goes to live in the country and sends her eldest daughters to Richard's court. She wasn't a shy and retiring wall flower - if she thought her boys were missing she would have kicked up a stink, and if she thought Richard had killed them she wouldn't have sent her daughters to his court - or come out of sanctuary for that matter, she would have feared for her own life.
As to the treatment of Perkin Warbeck... I can see the argument that Henry isn't sure who he is in his treatment of him. But I can also see that he knows full well that this man isn't Richard, Duke of York - but he can't admit to that without admitting that he knows Richard is dead - and how he knows that for a fact. After all - the way he got rid of Lambert Simnel was by producing the real Edward of Warwick. His treatment of Warbeck only proves that he doesn't have Richard of York to hand to produce - but that could be because he genuinely doesn't know where he is ... or because he know's he murdered him, but he can't really just come out and say that!
After all - if he hadn't repealed the Titulus Regulus then Richard of York would have had no claim to the throne anyway. But he did repeal it - thus making Edward V king and Richard of York his heir. He really backed himself into a corner and needed to sort it out. Having the boys killed would seem like a sensible option. And he sent their mother away to live in a convent, stripping her of all her titles and wealth, right around the time his own first heir is born. Almost as if he is trying to get rid of the one person who will cause a fuss if she notices she isn't being given news of her boys.
The boys are dead - Elizabeth is locked away - everything seems sorted.
...and then Perkin Warbeck turns up. He can't produce Richard the way he produced Edward - and he can't produce Richard's body without causing a scandal - so he fudges along until he eventually has enough to execute the impostor without having to come out and say once and for all that this is definitely an impostor.