Does Sherlock Holmes Really Want to Capture Moriarty?

In some ways no. Why do I say that? I suspect for Holmes, Moriarty is like another one of the narcotics Holmes uses. Moriarty is the most interesting criminal Holmes has ever dealt with. Holmes takes the drugs when his mind is not occupied. In that sense, Holmes needs Moriarty!

Holmes must eventually catch Moriarty in order to not be seen as having failed. It reminds me of the Gilbert and Sullivan Operetta, the Mikado when Katisha thinks the executioner has killed Nanki Poo and she cries "Where shall I find another?" In Holmes' mind, he will never meet another challenge like Moriarty.

Did Holmes want to kill Moriarty? No! Moriarty's going for that fatal swim is not what Holmes wanted at all. Holmes wanted to be able to bring Moriarty back to England in shackles and deliver him to British Justice. Holmes wanted to be able to see Moriarty tried on a capital charge and then be able to attend his hanging.

Moriarty certainly did not want to die at the Falls. If a camera could have followed him like a bungee jump, he probably would have thought before he blacked out, better this than Holmes taking him back Britain, humiliating him, and then having hated British justice finish him off.

I've read about many Serial and Spree killers such as Andrew Cunanan or Herbert Baumeister who will kill themselves to avoid capture. This way, they are in control, even to the end. If Moriarty were cornered, rather than shoved over the falls, might he have taken his own life instead, with his own version of falling on his sword? Food for thought.